That Rome was considered a Republic during the accession to power of Octavian/Augustus and quite for some time afterward was an artifact of Augustus' pure political genius. Primus inter pares he was most certainly not, and if the fiction was maintained by mutual agreement it was still fiction. By the time Caligula popped up it was no longer even that.
The Roman ideals popularized by such authors as Livy and Machiavelli were changing cultural norms spaced out over a period of around 500 years; one suspects that some at least are as "idealized" and distilled as the story of Washington and the cherry tree: illustrative but not entirely historical. And yet some of them - Cincinatus, Horatio at the bridge, a host of others - were not merely heroic but acted upon. One can dismiss either too little or too much in this sort of characterization.
But it's human history: an imperfect monarchy giving way to in imperfect Republic giving way to an imperfect Empire that was in reality the monarchy redux. If we're careful with the parallels we can learn a lot from them.
Good synopsis of the transition from the Roman republic to empire,
“That Rome was considered a Republic during the accession to power of Octavian/Augustus and quite for some time afterward was an artifact of Augustus’ pure political genius. Primus inter pares he was most certainly not, and if the fiction was maintained by mutual agreement it was still fiction. By the time Caligula popped up it was no longer even that.”
At what point would you say that the United States of America ceased to be a republic?