It was very effective. There are several examples of its effectiveness, particularly in the many battles where European forces expanded or consolidated their empires.

In the 1870s and 1880s the British used the gatling gun against African tribes armed primarily with spears in the Anglo-Zulu and Anglo-Ashanti wars (specifically the 3rd Anglo-Ashanti war in this case) and against forces with firearms and canons in the Anglo-Egyptian war (1882). In the battle of Tel-el-kabir (Anglo-Egyptian war) for example, the British managed a 50:1 ratio in Egyptian dead vs. their dead (2000 to 39), partially due to the use of gatling guns. In the Battle of Ulundi (Anglo-Zulu war, 1879) the British were outnumbered nearly 3:1 but through the use of two gatling guns and canons firing cannister shot were able to win the battle with a huge mismatch in fatalities (fewer than 20 killed on the British side, nearly 500 dead on the Zulu side).

In a particularly noteworthy example there is the Battle of San Juan Hill during the Spanish-American War (1898). While assaulting Kettle Hill 3 gatling guns were employed to provide continuous suppressive fire against the Spanish, allowing American forces to advance up the hill in what would have otherwise been suicidal conditions. In a period of about 8 and a half minutes three guns expended about 18,000 rounds, indicating the gatlings were in near continuous use during that period of time. After the Americans had taken the hill and while the gatling guns were being repositioned to the top of San Juan ridge a hasty counter-attack by about 600 Spanish soldiers was cut down with fire from just a single gatling, killing nearly the entirety of the force in a matter of minutes.

There are many other contemporary examples. Suffice it to say that the gatling gun was quite effective under the right conditions and effectively destroyed the potential for massed frontal assaults (even with favorable numbers of soldiers). That doesn't mean it was an unstoppable devastating force in all cases, it had its limitations just as any weapon does, but it was quite devastating in the few specific cases where it could shine (open ground, large numbers of close packed enemy troops).

The impact it had on warfare was to make frontal charges much less effective. It and its successors (the Maxim gun and other fully automatic machine guns) drove warfare increasingly toward the WWI model: trenches and fortifications across a front-line that was in most cases suicide to attempt to cross, even en masse. And that eventually led to modern combined arms tactics, the employment of artillery, small arms, light machine guns, grenades, mortars, mobile armor (tanks, APCs, AFVs, etc.), and even aircraft in a carefully coordinated assault. And, incidentally, the demise of cavalry (though artillery is also a significant contributing factor there).

The Gatling did have one massive problem and that was with Hanging ammunition.

When black powder ammunition has been stored for too long or exposed to moisture, you can get what's known as a Hangfire. This is where, after the firing pin strikes the primer, the round doesn't fire immediately, but smoulders before finally going off. The time between pulling the trigger and the round going off is called the Hang. Hangfires were common, especially were ammo was stored poorly or handled roughly (like on British campaigns in Africa) with most hangs being only a tiny fraction of a second. However, occasionally some can be much longer with some rounds never going off at all.

For a single shot rifle like a Trapdoor Springfield or Martini Henry, hangfires are not a problem, you just aim at the target, pull the trigger and wait till the round fires. If it doesn't, you give it a few seconds for safety, eject the dud round and load a new one.

For the Gatling Gun, hangfires were a massive issue. With a Gatling Gun, how long each round spends in the chamber is dependent on it's rate of fire which is controlled by how fast you spin the handle. Spin the handle fast, the gun fires fast but each round only spends a very short time in the chamber. The problem is, if the round is a hang fire, and it's not given enough time in the chamber to fire before it's ejected, it will go off inside the action of the Gatling Gun. As the round explodes inside the action, the two parts will fly away with equal energy, the lighter case will have much higher velocity, deforming and jamming the action. The heavier projectile will have much less velocity (same energy) and will either fly off into the action too or part way up the barrel. Whatever happens, a hangfire going off inside the action but outside the chamber almost always resulted in a jammed gun that was out for the duration off the battle.

To counter hangfires, British Gatling Gunners were trained to fire at two speeds. The first was a very slow rate of fire of only a couple of hundred rounds a minute. This gave each round the maximum amount of time in the chamber before ejection and limited the chance of a catastrophic jam. The second rate was for desperate circumstances when the line was going to be over-run but even then was not that fast. British Gatling Gunners were drilled endlessly so they would be able to maintain these two exact rates of fire in battle when the bullets where flying and their mates dying around them. But even then, the British found that the gunners couldn't help themselves as the wall of African spears came close, and a faster rate of fire led to a much higher chance of a jam, and a jammed Gatling in these battles often lead to annihilation.

To quote Sir Henry Newbolt 'The Gatling's jammed and the colonel dead, And the regiment blind with dust and smoke. The river of death has brimmed his banks'

The answer of course was to have the machinegun use the recoil of the round firing to eject it and load a new one. It didn't matter how long the round hung in the camber of a Maxim gun as nothing would happen until it fired which caused it to cycle. But today, with hangfires no longer being a problem, the Gatling Gun is back.

The main thrust of your question is about the Gatling Gun's effectiveness, but specifically in regards to the "The Last Samurai", one way that the movie overplays the power of the Gatling Gun is by putting it up against a bunch of samurai in armour with swords. In the actual historical Satsuma Rebellion, the Imperial army's superior artillery, including the Gatling, were indeed helpful in putting down the less well-armed rebellion, but the rebels were modernized soldiers who fought with modern arms, not swords, including Imperial artillery they had seized.