We really have no clue how it worked. If anything it probably was more related to being a head of a prominent clan/family.
The periods 293 through 219 BC and 166 to 9 BC are vague because of missing Livy volumes and lesser (or no) alternative sources for those ranges.
Aside from age, being a (primary major family) Patrician was a factor for some of the Roman Kingdom, at other times the pool expanded to include minor Patrician families. During the Kingdom the Senate largely worked as an advisory council to the king, grinding through legislative details, and more or less being responsible for the election of new kings (variously with or without input from "the people").
> Patrician was a factor
We don’t really understand the patrician vs plebeian split and how it functioned before or during the founding of the republic. Again there is evidence it was only fully established decades later.
There were some 700+ years of Rome before they occupied Britain and lasted for longer than the current age of the US senate system after that occupation.
During that long arc Roman senate ages, prerequisites and duties changed multiple times.
As for Livy, sure, like many historians attempting to cover several hundred years of events there are biases and myths laid as fact, etc. Even the events of the last twelve ears aren't agreed upon by all despite petabytes of live video and digitised records drilling down to the sale of individual shoelaces.