People have a tendency to romanticize the past and in terms of the Republic, the Romans were no different. Livy, in his histories of Early Rome, depicted Brutus (one of the men responsible for leading the effort of expelling the Kings) as making the Roman people swear a sacred oath to never let a king sit the Palatine again. As you may know, after the sack of Rome by Brennus (and the destruction of many records), really all the history is not set in stone. Livy's understanding of events could be completely wrong, however, what it does show that a good deal of people thought that the Roman people had sworn to never accept a tyrant.

Another important factor is to look at the emperors themselves, especially Augustus because he was a trend setter. In The Res Gestae Divi Augusti (The Achievements of the Divine Augustus - Brunt and Moore Translation) it is crucial to look at his word choice.

"At the age of nineteen on my own responsibility and at my own expense I raised an army, with which I successfully championed the liberty of the republic when it was pressed by the tyranny of a faction."

"The dictatorship was offered to me by both senate and people in my absence... but I refused it."

As you can see, even the leaders of the Empire were dressing themselves in republican garb. Despite the fact that Octavian had just waged a bloody civil war and annihilated the competition (some of which, arguably, was the legitimate Republic government). He was also the heir and adopted son of a man who had also waged war on the Republic and declared himself dictator (a temporary position) for life. Despite the obvious truth of Augstus' power, many wholeheartedly accepted his charade as champion of the Senate. Unfortunately, later rulers had all of Augustus' need for power, but none of his tact to mask their crowns. The failings and flaws of Augustus' immediate successors would only strengthen the cult worship of the dead Republic and inspire the elite to dream of the greener grass and those fabled golden years.

See here how Seneca depicts Cato the Younger, one of the key proponents of the Republic who stayed the course even until its demise:

“I do not know what nobler sight Jupiter could find on earth, should he wish to turn his attention there, than the spectacle of Cato, after his cause had already been shattered more than once, nevertheless standing erect amid the ruins of the Republic.” (Seneca, De providentia 2.9)

So it is fair to say the Republicans will still being fondly remembered even till the closing of the Julio-Claudian Dynasty.

It is important, as you have mentioned, to realize that this longing for the Republic is really only traceable in our sources because of the nature of our sources (i.e. rich, educated elite who had everything to lose from the fall of the Republic). We must be careful to realize this, lest we repeat the mistakes of our sources and improperly romanticize a system that had its own share of failings.

This Republican worship, and here is where this post turns into outright conjecture and theory, transcended praise of the individuals and worked its way in the collective Roman imagination. Rome's fascination with their defeated peoples it just one iteration of this concept. I believe that Romans (the intellectual elite that is) began to see more of their republican virtues in the actions and words of the peoples they encountered and fought, rather than in their own tacit acceptance of an autocracy that they swore they'd never accept.

Ch. Wirszubski, in his work Libertas as a Political Idea in the Late Republic and Early Principate, did a wonderful job fleshing out this notion:

"Tacitus thought the wretched state in which under the Principate the Romans repeatedly found themselves was of their own doing no less than of the emperor’s. The Romans themselves ‘rushed into slavery’; they abandoned their dignity and viewed with each other in contemptible adulation and abject servility. It seems the intense gloom of the Annals, in so far as it is not due to the nature of the subject matter, is born of the realization that the moral degeneration of the Romans is just as perilous as the worst despotism. indeed, what makes despotism. Indeed, what makes despotism so dreadful is the fact that it can stimulate and release the worst potentialities of men whose moral standards are gone." (p.164)

The consensus, at least in Tacitus' mind, seemed to be that the vast majority of men of character and worth died in the Civil Wars and Rome was simply left with the cowardly and degenerate. Compare this observation to the fact that some of the most stirring speeches of all time were put in the mouths of Rome's subject peoples by Roman authors. Tacitus made his depiction of the Caledonian chieftain, Calgacus, utter the famous lines: "To ravage, to slaughter, to usurp under false titles, they call empire; and where they make a desert, they call it peace." Another example is Josephus' retelling of the speech of Eleazar at Masada:

"And I think it is God who has given us this privilege, that we can die nobly and as free men, unlike others who were unexpectedly defeated... We are free to choose an honourable death with our loved ones… One thing only let us spare - our store of food: it will bear witness when we are dead to the fact we are perished, not through want but because, as we resolved at the beginning, we chose death rather than slavery."

This "worship" of the conquered, at least in my mind, pairs nicely with the concept that Romans saw their fabled republican virtues in the words and tactics of their inferiors, whereas when it came time to look at themselves they could only find those virtues in the actions of long dead ancestors.

As far as I know, although these feelings were certainly present, I am not aware of any instances where they coalesced into a meaningful, organized, or even active effort to return the Republic.