First, a couple of small clarifications. There was a moment on the assassination of Caligula that the senate contemplated doing without a Julio-Claudian princeps. Cassius Dio says:

After the murder of Gaius the consuls despatched guards to every part of the city and convened the senate on the Capitol, where many and diverse opinions were expressed; for some favoured a democracy, some a monarchy, and some were for choosing one man, and some another. In consequence they spent the rest of the day and the whole night without accomplishing anything.

Suetonius some time earlier said:

The consuls with the senate and the city cohorts had taken possession of the Forum and the Capitol, resolved on maintaining the public liberty.

Just for clarification, "public liberty" there is a translation of "communem libertatem." It's not clear that that means re-instituting popular elections of magistrates or the use of the assemblies to pass laws, but I think it's pretty clear that it means doing without an inherited princeps.

In the event, the senate conceded to pressure from the Praetorian Guard, who had selected Claudius.

Also, Domitian used "dominus et deus" - master and god - well before the 3rd Century Crisis.

Back to the Republic part of the question: You mentioned "restoring democracy." There is no sense in which the republic should be considered "democratic" in the sense that 5th century Athens is considered "democratic." In Athens, all male citizens had an equal vote, while at Rome the structure of the voting assemblies minimized the influence of the bulk of the people, to the extent that if the two richest classes voted together in an election in the Centuriate Assembly, they could win against all the other wealth classes. We're looking very much at an plutocracy of if not the 1%, a very small %.

Moreover, it doesn't seem like much of a "democratic" theory of the Republic was a thing. We don't have a lot of Roman political theory, but we have some. Cicero defines the Republic for us in de Re Publica (he puts this in the voice of Scipio Aemilianus):

Well then, a republic is the property of the public. But a public is not every kind of human gathering, congregating in any manner, but a numerous gathering brought together by legal consent and community of interest.

We get nothing here about popular elections or equality of voting or any of the things we associate with a modern Republic. Even further, Cicero dates the foundation of the Republic to the foundation of the city by Romulus - so during the monarchy, before the establishment of the voting assemblies.

The Orthodox view is very much that the Republic ended sometime around Augustus, though people quibble over the precise date. 31, 27, 23, are all common numbers people through out. For instance, here's the beginning of Syme's seminal The Roman Revolution:

The greatest of the Roman historians began his Annals with the accession to the Principate of Tiberius, stepson and son by adoption of Augustus, consort in his powers. Not until that day was the funeral of the Free State consummated in solemn and legal ceremony. The corpse had long been dead. In common usage the reign of Augustus is regarded as the foundation of the Roman Empire. The era may be variously computed, from the winning of sole power by the last of the dynasts through the War of Actium, from the ostensible restoration of the Republic in 27 B.C. or from the new act of settlement four years later, which was final and permanent.

So you have the Free State killed during the reign of Augustus and finally buried by Tiberius becoming princeps after Augustus' death.

Syme is drawing on Tacitus for this opinion. Here's Tacitus from near the beginning of the Annals:

When the killing of Brutus and Cassius had disarmed the Republic; when Pompey had been crushed in Sicily, and, with Lepidus thrown aside and Antony slain, even the Julian party was leaderless but for the Caesar; after laying down his triumviral title and proclaiming himself a simple consul content with tribunician authority to safeguard the commons, he first conciliated the army by gratuities, the populace by cheapened corn, the world by the amenities of peace, then step by step began to make his ascent and to unite in his own person the functions of the senate, the magistracy, and the legislature. Opposition there was none: the boldest spirits had succumbed on stricken fields or by proscription-lists; while the rest of the nobility found a cheerful acceptance of slavery the smoothest road to wealth and office, and, as they had thriven on revolution, stood now for the new order and safety in preference to the old order and adventure.

That is, to the best of my knowledge, the first time the claim that the Republic died with the rise of Augustus occurs in extant Roman literature. The historian Velleius Paterculus, writing under Tiberius, who knew Tiberius and fought with Tiberius, had written:

The old traditional form of the republic was restored. Agriculture returned to the fields, respect to religion, to mankind freedom from anxiety, and to each citizen his property rights were now assured; old laws were usefully emended, and new laws passed for the general good; the revision of the senate, while not too drastic, was not lacking in severity.

He thought the Republic was trucking along just fine under Tiberius. You know Augustus also claimed to have restored the Republic. Tacitus, I suspect, was referring to Velleius when he talked about "Hence my purpose is to relate a few facts about Augustus... without either bitterness or partiality, from any motives to which I am far removed." Paterculus was very much a fan of Tiberius, but that doesn't mean he's a shill, only that his perspective is different from Tacitus'.

All of this is to say that it's not clear that anyone contemporary with Augustus thought his rule marked the end of the Republic; even when the senate was contemplating going forward without a Julio-Claudian princeps it isn't clear what they thought they meant - our sources for that are all 2nd C CE or later. Even Tacitus on that point is lost. The notion that Augustus marked the end of the Republic only pops up with Tacitus, writing under the Flavians, after no Julio-Claudians were left.

Syme was probably the most influential Roman historian of the 20th century, and I suspect his propagation of Tacitus' point of view has a lot to do with that narrative being the dominant one. For instance, in the mid 19th century you find paintings like this, in which Gerome celebrated the reign of Augustus as one of universal peace comparable to the reign of Christ, and used it as an analogy to celebrate the ascension of Napoleon III to the throne of France.

That's a little bit rambling. Let me know if I need to clear anything up.