They were extremely, ridiculously, irreplaceably important. What a lot of people have is an unfortunate layman's understanding behind the science of warfare, exacerbated by films, TV, and books that confuse and muddy what it is a general actually does at the head of his troops.

Here's the scenario you are most likely envisioning:

Two armies stand on a battlefield. They've probably found small slopes for the generals stand on so that they can achieve a commanding view of the field with their host of men arrayed in front of them. The general, who has led his men to this point to achieve his strategic victory, sits atop a horse surrounded by bodyguards and bellows out commands.

"Archers," he calls out, "knock." Then as the enemy gets just close enough he tells them to loose. When the bombardment has finished he waves a hand and orders his soldiers back, then gestures and calls out for spearmen. But wait, the spearmen are surprised by a sudden flank, so he wheels around and calls out for his cavalry to come in and cover them.

Huzzah the day is won!

The issue of course is that this isn't what a general does at all and though they did their best to keep a bird's eye view of the battle, for a million reasons he is not going to be running around bellowing out specific, tactical updates that all men will hear and recognize. Before I get too deep into that, let me just tell you that the reason he won't is because no one is going to hear him anyway. Battles are loud, noisy, chaotic things, and no matter how commanding a presence he is, he's just a guy on a horse.

What a general actually does is get his army into a position where they achieve victory. This means he's doing a lot of things that don't make for terribly interesting TV, but are fascinating to learn about all the same. They are the managers, the supervisors of their army and this takes up the absolute majority of their time.

The general in command of an army will always have a number of absolute priorities that come before anything else at all:

1. His men are fed
2. His men are supplied
3. His men are present
4. His men are moving on an objective

Before a general ever even considers addressing any possible issue those are his absolute four priorities. Now you may be asking, "Isn't fed and supplied the same thing?" No, not at all. Armies are entire small towns moving about and all the things that requires.

We'll look at the obvious ones first. Men need shoes, men need armor, men need weapons, men need ammunition. But they need more than that, an army doesn't survive on bare bones.

Your army needs raw materials such as iron, it needs extra clothes, it needs blankets, it needs leisure, it needs coinage to pay its soldiers, it needs paper/parchment, it needs wood, it needs wax, it needs horses, it needs hay for those horses, it needs extra plumeage for the officers to wear on their helmets, it needs mail to go back and forth between the supply lines, it needs pillows and livestock, it needs hammers and medicine and wool and cloth and the list goes on and on and on.

And the general has to provide for all of that. He has to make sure that all of these things are coming in, then he has to make sure that the men are marching the amount of miles that they need to march, that they're well rested and morale is high, that they're in proper fighting shape. Sure your men are walking, but are they eating enough? Are they drilling constantly? Are they maintaining their equipment?

The general is finding out where they are getting food. Is it about the huge supply line following them, or are they foraging and pillaging?

Then the general is insuring that discipline is being maintained. Men are respecting their officers sure, that's obvious. Are men sleeping too late? Are they complaining too much? Are they listening to orders? Are they performing their camp functions? Are they saying their prayers or honors of ceremony? Are they wearing their armor or uniforms with distinction?

Then the general maps out their march. How many miles can they cover each day? Are you pushing your men too hard? Are you pushing them hard enough? What direction are you taking? What roads? How many hours of the day?

Now the general is making these decisions because he's constantly communicating, all day every day, with dozens of officers. He's reading scouts' reports, he's listening to information about the weather, the quartermaster and his staff are communicating the levels of food and goods and how much makes it to your army. Your cavalry commander is telling you about the status of the horses, how many horses have gone lame, which ones had accidents, bucked and threw a rider to the ground. Your various infantry commanders are telling you about desertions and energy levels.

When you're done with all this you are in communication with "home base", whatever that may be, so that they can continue to operate their grand war with as much up to date information as possible. You're altering requests for new men, new supplies, you're telling them your destination and your ETA, you're accepting new orders and changing your plans around that.

And think for a moment, we're just talking about an army marching from Ravenna to Narbo, through friendly Roman territory. We have yet to even address what happens when you venture into enemy territory, and you're dealing with hostile scouts, raiding parties, and threats to your supply lines.

But still before you go into hostile territory you are taking the barest principle that your government, city, kingdom what have you can give you, and you're condensing that into a proper objective. You have been told to end the rebellion blossoming around Transalpine Gaul. You now need to communicate with people who have information necessary to be converted into intelligence so you can decide how to achieve that objective.

Therefore you talk to veteran officers who have served in the area, you talk to spies and locals. You find out what cities and villages are important to the rebels, you find out where their food comes from, their level of military sophistication.

Okay so knowing all that you know that the best way to bring down the rebellion is to bring down Massilia. Your scouts tell you that the Gallic Warlord leading the rebellion relies on sustaining himself through foraging, but also threatens Roman allied tribes to extort food and materials out of them. Now you have to decide if you will just march straight to Massilia and try to draw him out or if your strategy will involve defending the tribes that are being threatened and hope to catch him in the act.

Okay now you have your strategy, now you drill your command staff and pick lesser officers based on a mixture between how to best placate the senators who have released the money to finance your campaign, or based on their expertise in fighting Transalpine Gallic Warriors.

You set out on your march with all these things constantly going on, reacting to updates as they come to you in limited, uncertain information, doing your best to make sure that when you left Mediolanium in the early Spring with 10,000 legionaries and 2,000 auxilia you arrived in Gaul with at least most of them.

Then you finally do battle, drawing up battle plans as you meticulously read over scouting reports of dubious quality and discuss the best course of action with your staff. Now comes the fun part. Remember you're not just Glorious Imperial Commander who speaks ex cathedra. You are a personnel manager for several department heads. You break up arguments and try to keep the hot headed cavalry commanders from crushing morale by lording their aristocratic status over the grizzled old infantry commander you've come to rely on. You have to mediate disputes as conflicting personalities are constantly damaging morale and pride is getting in the way of everything. You want to befriend the cavalry commander, but he's haughty and arrogant and wants you to approve his plan, but it's obviously a disastrous one. You fear he won't have your back at some critical junction later.

Then you fight and hopefully you win, after thousands of your employees and friends lie maimed and dead on the battlefield, leaving you wondering if you could have done something different to get to this point.

And to think, I've only barely brushed the surface without going into detail about the meticulous way you choose officers, how you handle post victory policies, how you use this victory to escalate a political career, how you save yourself from enemies despite your victory, how you plan for a defeat. I mean this list just keeps going on and on.

Without a general and his officers a legion is 6,000 people in iron walking around the country side.

Caesar may have had his legions, but so did Crassus.

I'm going to speak only in my expertise (Ancient China) but I'm pretty sure this applies to a lot of the generals of the ancient world. In Ancient China, generals were powerful people in both warfare and politics. It was critical to any Dynasties success that you had to have able generals who knew strategy (Sun Tzu's Art of War was very prominent). Once a general had to march an army, they were assigned other generals who they were in charge of and they often split into multiple armies. It was also common to be assigned some strategists or advisers who were knowledgeable of key factors such as weather, geography and even battle strategy.

It was up to the commander of the army to decide battle strategy, tactics, when they would attack, if they would withdraw etc. Many generals became famous for their prowess and tactical ability on the field (Xiang Yu, Cao Cao, Guan Yu etc). Through victories, generals gained more and more fame, higher positions in the political spectrum and were often granted posts in cities where they would become prefects and would often lead out armies themselves if their city was attacked.

While there's no doubt that sometimes victory was achieved because of superior numbers or a better army (which you can credit to the generals for having such superior trained troops), that wasn't always the case in Ancient China. Cao Cao who was probably the most successful leader during the Three Kingdoms period (220-280), was able to defeat his bitter rival Yuan Shao at the battle of Guandu despite being heavily outnumbered. In contrast, Cao Cao was also defeated at the battle of Chibi despite having vastly superior numbers due to the wit of Zhou Yu. Another example is Xiang Yu defeating his rival Liu Bang with a significantly smaller army at Pengcheng.

I apologize if this is a little sloppy, but I hope I shed some light on your question.

tl;dr: Yes, generals were hugely important in warfare and even politics.

I'll answer for the realm of history you are asking about specifically.

Generals were really important in a few ways, strategic leadership was only a small case. Their most important role was as a symbol in their army. This is especially the case with Alexander. He led the cavalry charges personally (I think it's fair to say that Alex's success was due more to his cavalry than his phalanx) and suffered in the desert along with his troops instead of taking water when it was offered. These examples became the goals of Hellenistic and Roman monarchs long after Alexander. They were expected to have military success and lead the army themselves. So you can fairly say that the role of general was important, even if their tactical decisions were not the deciding factors in a battle.

In republican Rome the case was a little different since you could expect one consul to replace another with little harm (you could say the same about Athenian generals too, I suppose but they were actually elected unlike other Athenian magistrates and could serve more than one year at a time). Roman generals usually kept command of their armies until their mission was done even though they lost their political authority after one year in office. However, they were often exchanged. Scipio Aemilianus took command of the Numidian War in 134 BC and Pompey replaced Lucullus in the Mithridatic war in 63 BC for example.

These practices show two conflicting realities of the Roman military. On the one hand, it must function well no matter who led it because their leader could be changed at any time. This was especially true of armies in provinces like Spain where the military was never actually sent home. Thus soldiers, instead of going home with their commander as Roman armies usually did, would be left to serve under the next commander since levying men to serve abroad was prohibitive.

On the other hand you have the idea that changing a commander will have an effect on the outcome of war. Clearly the Romans understand that some commanders were more successful than others. Scipio Africanus was able to defeat Hannibal in open battle on Hannibal's home turf when Hannibal defeated larger Roman forces in Italy three times! Marius was able to defeat the Cimbri and Teutones by reforming the military after his predecessors had allowed the Germans to cross the Alps into Liguria. These examples should be enough to show that even though the Roman military had great logistical and tactical traditions, it sometimes took a man of strategic and tactical vision to actually make the legions work. Marching in a quincunx (the standard checkerboard formation of a Roman legion) worked for many Roman commanders in many wars, but it did not work against Hannibal. Hence while the army was excellent, it also needed a general who knew when tradition and training was not the tactical answer.

Another Roman tradition called the spolia oprima shows how important the enemy general was on a cultural level. The spolia oprima was the act of stripping the enemy commander of his armor in the midst of battle after single combat. Think of the scrambling over fallen heroes in the Iliad. Only a handful of Romans ever claimed the spolia oprima, but historians, statesmen and poets recalled them often. The fact that taking a commander's weapons in battle was so memorable shows how important the Romans saw enemy leaders and, by reflection, their own.

I've seen it argued that Alexander's victories were due to the superiority of his phalanxes

Sigh. This is why I don't care for Dan Carlin. He was the first person I saw to widely circulate this statement and frankly it's full of crap. To believe this you have to ignore the fact that Phillip of Macedon also had the macedonian phalanx (He invented it after all) and he was not nearly as successful as Alexander was. He struggled for many years to control, greece, and while he was successful, he did not have the blazing success that Alexander did. Look at any of Alexander's great battle, in each of them he demonstrated a capacity for deception, timing and aggression. While conquering the Persian empire, his logistical skill allowed him to cross enormous distances with relatively little attrition. In his march across Iran, and to Egypt in particular, he crossed deserts that destroyed lesser armies with worse logistics. The battle of Jaxartes, Gaugamela and Hydaspes all clearly demonstrate his superior visualization of the battle space and ability to direct his forces with good timing. The Siege of Sogdian Rock is a perfect example of Alexander seizing a fortress that defied almost every other commander in history.

I also feel that Hannibal's reputation is justly deserved. The battle of Cannae is a master piece of playing to your strengths, and delaying the enemy. I feel that the Battle of Trebia is another clear example of this.

This is especially absurd in my own area of interest, early modern warfare in europe, 1500-1700. For example, Gustavus Adolphus revolutionized how battles were fought on the tactical scale. He introduced the concept of volley fire for shock effect. He also reorganized his cavalry wings, and enforced higher drilling standards resulting in better performance from his army. All of these changes were copied by the other armies of his time.

I could go on with Wallenstein, Pappenheim, Turenne, Conde, Torstenson, Vauban, Marlborough, Frederick the great, Napoleon, and Moltke. Each of these commanders had a profound affect on the way their respective armies fought battles. I feel that a lot of confusion comes from the fact that just because a commander wins all the battles, that doesn't mean they win the war. Hannibal and Napoleon are the poster children here. This occurs when one side has huge logistical shortcomings compared to their opponent despite large tactical advantages. For Hannibal, the Roman empire was larger, richer and better organized than Carthage. Thus Hannibal won dozens of battle but couldn't win the war, while Rome only had to win a single battle. Napoleon was in the same boat. He was trying to conquer Europe and Russia with the resources of France. An impossible task. If anything, these cases show just how big an effect a competent commander can have.

No, they had a clear vision the means with which to execute that vision and the ability to carry their arms to victory they were absolutely the geniuses they're made out to be. Rome's legions were hardly invincible even with excellent commanders to lead them, the Battle of Gergovia being a prime example from Caesar's own campaigns. Phalanxes also have a mixed track record in terms of effectiveness and are only exemplary in certain situations and terrains. Hannibal's cavalry was superb, but it was far from some type of overwhelming advantage that made his army unbeatable. Instead it was these commanders supremely gifted ability to identify their army's advantages and enemies weaknesses and exploit them. That is not to degrade the supreme organization and discipline of the legions or the uses of the phalanx, but in the hands of lesser generals they fail to accomplish what Alexander or Caesar accomplish. Hannibal's ability to hold together an ethnically diverse army for over a decade on the Italian peninsula is not a testament to his cavalry. His battles at Trebia, Trasimeine, and Cannae had as much to do with his infantry's prowess as his cavalry's.