Senators were not elected, nominally or not. In the very earliest period senators appear basically to have been appointed for life by the consuls, but by the lex Ovinia of 312 the censors were given the power to choose senate members in the lectio senatus. Mostly that meant that curule magistrates and the occasional quaestor or aedile was automatically coopted into the senate, although occasionally people could become senators without holding major offices. The Scipios were particularly guilty of this, and under Cato and others in the second century their hold over the state, which was much tighter than I think a lot of people realize, was opposed. Among the points used against the Scipios and others like them was the institution of minimum ages for magistracies, the first step towards a codified cursus honorum, starting with the quaestorship by 180 at the latest. By 123 plebeian as well as curule aediles took senatorial seats after their terms of office, and Sulla opened the senate automatically to ex-quaestors. By the middle of the second century extraordinary cooptions into the senate basically stopped happening. The magistracies that, upon the end of the year of office, granted access to the senate were elected, but senators were not.

However, this is all beside the point. The senate did not "justify its own rule." The senate did not "rule" Rome, nor did it claim to in the Republic. Foreign envoys, either from other states or the provinces, frequently referred only to the senate when speaking of the leadership of Rome, but that was because envoys delivered their speeches in the senate, not before the people. Amid all the rhetoric we find in Cicero and other Republican authors, not only in speeches ad senatum but also ad populum, about how the best men should have the most influence and how the infamia plebs is an untrustworthy group that should not be listened to, there is nothing about the senate ruling Rome. There isn't really anything about the magistrates ruling Rome. Quite the opposite, actually. Republican speakers appear to have routinely spoken of the magistrates as doing the bidding of both senate and populus Romanus. This seems to have been a major point for both Sulla and Cinna, who rallied support by pointing out that it was not only their own rights as consuls that had been trampled on, but the right of the people to elect their own leaders, and that the people were being attacked, not themselves.

This is to say that yes, the Republic had a concept of popular sovereignty. It was the Republic's rhetoric about the populus on which the 17th century thinkers based their own thought, explicitly. In our earliest sketch of the Roman state Polybius describes three masters of the state: the consuls, the senate, and the people. All three, Polybius says, exercise powers that grant the powers of the other groups or take them away: the consuls are military leaders, the senate funds the consuls on campaign, and the people elect the consuls and decide by plebiscite whether the senate's suggestions are actually to be carried out. The extent to which Polybius' description reflects the Roman understanding of their own state or whether he's applying Greek constitutional theory is debated, but pretty much no living Roman historian of the Republic would dispute that some idea that the populus was the center of the state existed at Rome. It's everywhere in political rhetoric, and in political practice the assemblies were able to assert their authority in the face of senatorial opposition much more frequently than was long supposed--indeed, in the late Republic a majority of contested bills passed in the face of senatorial opposition. If we factor in the tribunes or look at contiones the model is only further reinforced.

The debate in Republican political history is not whether the populus as a concept was politically important. It was, there is no question. Rather, it's what exactly the populus constituted, and whether the populus, real or imagined, actually mattered. I've mentioned a point in the second part, namely that the assemblies were more often than not successful in counteracting senatorial decisions. To that I would actually also add the struggle in the first century between the senatus consultum and the lex, which we don't have much information for but which certainly existed--a senatus consultum, for example, of 64 that outlawed most types of collegia was overruled by a lex of 58, passed like all leges by the assembly. The first bit though is especially knotty. We know that the populus, for formal purposes, constituted whoever showed up to an assembly at any given date. For many routine actions of the assembly, like ratifying provincial commands, we probably shouldn't expect attendance to be especially high, and we know of a few cases where tribes had to be split so that part of the tribesmen could be assigned formally to a different tribe that had had no members attend the assembly. But at other times our evidence suggests that the assemblies were packed. Elections, which came at predetermined times of the year, and leges, which required nearly a month's advance debate before voting, often had huge attendances. The participatory nature of the Republic aside, what's certain and consistent across all these incidents is that the populus is being invoked constantly, by all sides and all those involved.