My answer is based on Politics in the Roman Republic from Henrik Mouritsen and The Beginnings of Rome from Tim Cornell, and to a lesser extent on Syme's Roman Revolution.

First, as you recognise in your question, the Senate was an incredibly long-lived institution, and as such it went through a lot of changes.

Let's start at the beginning. We know next to nothing of the age of the kings, and barely more about the early Republic. But, according to Cornell, and Mouritsen as well, the Senate was in the beginning merely a collection of advisors to the rulers of the state. While it did have informal power, as every advisory institution has some, it was not the government of Rome in the sense as it later became. It is probable that kings and early consuls/consular tribunes could choose the members of the Senate, and could also dismiss them if they wished so. Thus the composition of the Senate could change from year to year.

However, Republican Rome was a place where the wealthy elite ruled. In every kind of aristocratic government it is a paramount question to eliminate real competition between the oligarchs and find a "fair" way to distribute power and resources. That was presumably one of the driving factors why re-election of magistrates was curtailed over time. Mouritsen argues that this was mostly done by the ruling elite coming to an understanding, and applying peer pressure against those who stepped out of the line.

That does not mean that re-election of magistrates stopped altogether. In fact almost every generation has emblematic figures like Quintus Fabius Maximus Rullianus, or his great-grandson (or grandson) Verrucosus, on whom power was bestowed again and again. But your average Roman noble by the third century BC could only hope to become consul once in his life-time.

As consuls were changing practically every year, the need for a stable governmental organisation grew and grew. There was foreign policy to be made, complicated financial issues had to be solved, and the Senate slowly took over all these branches of the state. Nominally, the Senate had very little power. Magistrates were elected by the various assemblies, and also it was the place of assemblies to accept or reject laws. Likewise, the executive decisions were in the hands of the magistrates pro forma.

But the Senate had all the informal power. In fact almost no law came before the assemblies unless the senate already accepted it, and while the various magistrates still had very wide powers, they became more or less the agents carrying out the Senate's will. Mouritsen argues that elections and "referenda" were really legitimising acts only. He cites various examples when an election/vote didn't yield the desired result, so the magistrate who convened the assembly in question asked the electors to change their decision, and they dutifully did so.

I am not really sure that is 100% correct, and I don't understand if the votes were symbolical anyways why was the distribution of freedman in all 35 tribes such a controversial and longstanding issue. Or similarly why were the Italians after the Social War distributed in ten new tribes at first. But Mouritsen does make a convincing argument that elections and votes often had pre-determined results, and those results were ordained by the Senate.

The powers of the Senate were finally made formal by the dictator Sulla. Mouritsen argues that Sulla basically did nothing but legalised an already existing state of affairs. However, a few years after Sulla kicked the bucket Pompeius and Crassus committed their state-capture, and until the civil war against Caesar, Pompeius and his closest allies were more or less the shadow-government of Rome. Caesar, after he won the civil war held both formal and informal power in his hand, and side-lined the Senate.

The (second) Triumvirate also very ostentatiously concentrated legislative and executive power. Octavianus later reversed this: while he never gave up an ounce of power on his own accord, he was more than willing to pretend that he was not an absolute ruler of the state. Later emperors were often trying to imitate him, and conceal their rule with showcasing how the Senate and People were still running things. The performance was sometimes convincing, sometimes everything but.

However here I have to quote Syme, because it is really a sentence worth remembering: "In all ages, whatever the form and the name of the government, be it monarchy, republic, or democracy an oligarchy lurks behind the façade; and Roman history, Republican or Imperial, is the history of the governing class." (Page 7 in the Kindle version) The Senate as an institution had a reduced power under the Principate. But certain prominent nobles sitting in the Senate might have been still very powerful in their own right, and might have been still running the state behind the curtains, just like their forefathers did in the days of the Republic.

I hope this reply was at least somewhat coherent, and understandable.

One more thing, in your question, you also asked whether the people were happy with oligarchic rule. While the answer for that could go on at least as long as the previous part of my reply, I will say that conflicts between the different economic classes of Roman citizens can be seen through all of Roman history. In the early Republic there is the revolution of 450, later several other secessions of the plebs, and even after the Patrician-Plebeian struggles officially "ended" there were always movements pushing for land reforms, more equal distribution of the electorate in the tribes, and several other issues. But the Roman ruling class was always remarkably successful in presenting a united front towards these movements, and reach a compromise.

EDIT: Spelling