This is a great question! I am glad that you mentioned the armies of the late republic, as the situation is actually quite similar to that of the middle/late empire. The third century for Rome was an extremely tumultuous one, and is often called the "Crisis of the Third Century". It really began after the death of Marcus Aurelius and the ascension of his son Commodus to the position of emperor, and did not end until the establishment of the Tetrarchy by Diocletian.

During this time, Rome as a city had lost much of its power and influence. The emperors of the third century often did not even live in the city itself, and the senate had lost almost all legislative clout. The empire was very much under the sway of the army, as shown by the extremely rapid turnover of emperors during this time period. You mentioned privately funded legions during the late republic; much the same was going on during this period as well. Essentially, the soldiers would follow whoever would pay them. These payments were called "donatives", and consisted of influential generals paying each of their soldiers large sums of money in order for them to follow him. Cassius Dio, an ancient Roman historian, shows the use of donatives pretty well in his account of how Sulpicius became emperor (book 74, starting at sec. 1 with Pertinax).

Looking into it further, the soldiers were not really "fanatically loyal" to their general. Instead, it seems like they were more loyal to the money their generals could pay them. If you look at how most of the emperors during the third century died, you'll notice that many of them were murdered by the Praetorian Guard, the supposed "bodyguards" of the emperor.

This is one of my favorite periods of Roman history, so if you have any other questions, I would be more than happy to try and answer them!

I think there are some misconceptions about this question. I don't do high Empire stuff, but I do do Late Republic - 70. Let's start breaking this down.

I'm a first year student of history and currently doing some research on the Roman emperors during the Year of Four Emperors. One overarching theme of this crisis and other Roman civil wars that I've noticed is that generals who make a bid for the purple are proclaimed emperor by their troops before actually making their move. Moreover, they also seem to be able to quite easily mobilize the armies under their command to march on Rome and potentially fight their own countrymen.

Later in the Roman Empire this is true. Before 69 that wasn't the case. What happens in 69 is a sea change in Roman politics which I discussed a while ago here. TLDR Vespasian reformed the sources of legitimacy for Imperial rule, to be based on the law (lex) passed by the people - and one understanding of the Roman people was always the Roman people as the army.

What happened in 68-69 was quite a bit different. Nero was increasingly unpopular with the Senatorial class, but apparently very popular indeed among the people in Rome. The rebellion started with a Romanized Gaul Vindex, who tried to get Galba to revolt. Galba refused. Vindex was crushed. Nevertheless Nero and the Senate declared Galba a public enemy and basically forced Galba into open revolt.

Now Galba, in revolt out of desperation, started gaining support. Nero disappeared and in that moment a rumor started about his death, and the Senate agreed to appoint Galba Emperor. Who else were they going to choose?

This was a legit selection of a princeps by the Senate. It's the same way Tiberius and Claudius got their position. There was a complication because Nero wasn't dead yet, but he fixed that himself when he heard the Senate had selected a new emperor.

So I don't think it's fair here to speak of the army as the legitimating power behind Galba's ascension. It's almost Shakespeare-ish (Shakespearian?) in the miscommunications that were required to arrive at that conclusion.

Otho corrupted the Praetorian Guard to overthrow Galba partly with money, but also partly because Galba trusted him to run the PG. Vitellius claimed legitimacy by defending the choice of the Senate - Galba - so there are still real ideological claims to legitimacy beyond simple payment. Vespasian, as I mentioned, was the real shift in the source for imperial legitimacy.

Now I know that during the days of the Late Republic legions were often privately funded

Rather, during the days of the Late Republic legions were publicly funded usually, but depended on their general to get the government to pay for them.

Moreover, where the civil wars of the republic might have been somewhat confusing as to who was right and who was wrong, it seems to me that if you live in the empire you know exactly who is the lawful emperor and who is the usurper.

Why? Once you live in a government where legitimacy is determined by the "people", the question becomes who the "people" are. In the Republic this was well defined by the voting assemblies, but Tiberius abolished the voting assemblies. Who's to say who the people are then, except the people around yourself?

how is it that during the days of the Roman Empire, soldiers were so much more loyal/devoted to their generals than their country? I can imagine one or two insanely charismatic generals convincing all of their troops to fight and die for his personal ambition, but so many times?

I'm not an Islamic scholar, but I feel like the decades after the death of the Prophet might be a comparative example?