Octavian was not very good at winning battles (but good at getting others to win them for him), but had superior political skills. He had to avoid antagonizing the old senatorial families, but at the same time he needed to reassure the common people that he would remain in control. They didn't want a return to the free Republic which would mean more civil wars. So he had to work out a system by trial and error, pretending to "restore the Republic" while making sure he had the essentials of power. Having most of the legions under his command was a major component of his power. It was a military despotism but disguised in order to make it acceptable to the senatorial elite.
The ancient Greeks would have something pithy to say about that :)
[Speaking of the Greeks, they had the same cultural privileges that Britons or the French have vis-a-vis the American governing class. The Greeks often snooted the Romans of higher station, and those Romans tended to take it and adapt the customs of the snooters.]
Octavian was not very good at winning battles (but good at getting others to win them for him), but had superior political skills.
He definitely learned from the fate of his great-uncle. Also to his credit was his spartan habits. He had the reputation, even when Emperor, of living about as modestly as a private gentleman. That said he wasn't in it for the plunder, which said "trustworthy" to so many Romans.
You're right in that most Roman decided that the old Republic had rotted through. The ones who wanted the Republic restored also wanted the old-time virtues to be restored too, as they too recognized that the Republic had turned into little more than a corrupted husk at the time of the civil wars. Unity meant a lot, lot more to Romans than to Americans. Calling a fellow Roman 'un-Roman' was about as shocking and scandalous as a Roman consciously acting in an un-Roman way. Needless to say, Rome never had an Alcabiades, let alone an Alger Hiss.
If I recall correctly, there was a law that obliged the Roman judiciary, in any case between a Roman and a non-Roman, to side with the Roman. That kind of law enjoins, or at least encourages, a feeling of mutual unity. When foreigners had a gripe with Romans in the old Republic days, they sent emissaries to the Senate, which did have the authority to rule in foreigners' favour.
That law, and the underlying custom plus other jurisprudential consequences, meant that Rome was different from America in another way. When St. Paul said to the Roman who arrested him "I am a citizen," he got much better treatment. In fact, Roman rulers preferred to finance the State through foreign plunder over domestic taxation. As a result, Roman citizenship was prized until the near-end of the Western Empire. Had a Roman in Augustus' time heard about the Facebook co-founder who renounced his citizenship, plus the other tax exiles, he wouldn't have been shocked: he would have been plainly baffled. Especially about the economic benefits of self-exile.
It's hard to remember sometimes, but America ain't Rome. There are definite parallels - Prof. Madden's book Empires of Trust is an erudite look at them - but the two polities aren't the same.