I cannot recommend the History of Rome podcast highly enough, if you want to learn about the history of Rome.
You said, and I quote, "every patrician family had people in the senate and that’s who it was supposed to represent."
The only way every family would always have representatives would be by law. And that is not even going into the idea that the Senate represented the patrician families alone. It was the SPQR, the Senate of the Peoples of Rome, not the Senate of the Patricians of Rome. From the start, it represented the entirety of the population of Rome (unequally, I'll grant you, but the whole thing).
Grey Wolf