For example, Augustus's titles and attempts at claiming to be a princeps and the republic was still in place could plausibly be read in isolation as a republican system, and if he died soon after his seizure of power and the system went to something more resembling the republican system of politics, it might be remembered merely as a weird time with civil wars, dictators, and the like, but still in a system without a true king. When was an imperial system engrained strongly enough in Rome that even a return to a republican form of government in practice couldn't save it from being viewed in modern times as an imperial monarchy?
The legal position of the "Emperor" or "princeps" began to be codified when Tiberius took over. Until the powers of Augustus had passed to him without incident, especially the tribunicia potestas and the imperium maius, it was all just an experiment. Augustus' greatest contribution to the changeover was the fact that he "ruled" from 31 BCE to 14 CE -- 45 years, so long that most began to forget what the "Republic" had been like, and certainly there were none left alive in 14 CE who remembered a time before the decades of the two triumvirates. The "normal" Republic, if it ever even existed, had not been around since the 60s BCE, or even earlier.
At the accession of Caligula, the "position" was further standardized. By the time of Nero, second half of the 1st century CE, the role of the Emperor in the functioning of the state was becoming irreversible, or at least complicated to unravel. The Imperial finances (fiscus) vs the State Treasury (the aerarium) in particular became enmeshed in the running of the state. How soldiers were raised, deployed, and discharged. Et cetera. The reign of Vespasian is important, due to the actual legal codification of the lex de imperio Vespasiana. This document, though not complete, is partially preserved and seems to be an attempt to legally rectify the position of the Emperor within the "Republican" Roman state. From this point forward, the Emperor becomes more and more ingrained in the Roman state, until by the end of Antiquity, they are inseparable. When the Emperors begin to weaken and lose their relevance, especially in the west, all sorts of state functions begin to breakdown. By that time, many bureaucratic tasks are being overseen by Imperial freedmen (like maintaining the water supplies, distributing the grain dole, etc).