EDIT: While I love the period, I'm not an expert, so please read /u/LegalAction's counterpoints below.

There is an overly idealised fantasy, portrayed in for example the film Gladiator though it is not limited to Hollywood, of the Roman Republic as this freedom-loving democratic nation. It was not.

The Roman Republic was an oligarchy of an elite group of super-rich families, which were constantly competing with each other for prestige and power.

Wanting to go back to the Republic would only make sense if you were one of the elite super-rich families who wasn't the Emperor. In which case though you're more likely to just try and become Emperor instead.

I asked a question about this to one of the experts here recently. You can read the answer for yourself here. But the TL;DR is that for the majority of people all that changed was that it was now the Emperor instead of the Senate appointing their governors/high administrators. The systems below that remained the same, including various local republican systems.

In either case if a return to the Republic had happened it wouldn't have lasted. Augustus was actually not the first Roman to cease total power after winning a civil war, and neither was Caesar. For that you need to go back around 30 years to Sulla. Sulla actually tried to reform the republic so it would survive. But all that happened was that as soon as the generation after him came of age they fought their own civil war.