My knowledge of Republican and Imperial Rome is limited, and greatly influenced by Mike Duncan's History of Rome podcast. That being the case, it seems to me that the Roman Senate was always more concerned about preserving their own wealth and power rather than the welfare of its citizenry. Further, it was little more than a rubber stamping body during the imperial period.
Why did the founding fathers choose to emulate them and adopt its name?
Duncan's interpretation, I assume from your question, is essentially a very modern one. It isn't an invalid one, at least to the extent that one can attribute a guiding set of actions to a body as fractious and dysfunctional as the Senate (remember, Clodius was a Senator as well), but it is not how people like George Washington or Thomas Jefferson would have seen it. Even thinkers like Machiavelli, who was not terribly sympathetic to aristocratic interests, generally saw the Senate as a positive institution.
The Roman Senate was a fairly complicated beast. I don't know what you know from Duncan's podcast, so I'll keep it basic.
The word "Senate" (and Latin "Senatus") derives from the Latin word "senex", meaning "old man" - the Senate was, literally, a group of old men. It was raised from its original purpose of subservient administration under the king as an alternate to monarchy, following the expulsion of the hated Tarquinius Superbus by the legendary Brutus.
While the Senate was never held accountable to the everyday wishes of the people, the aristocratic Senators and regular citizens functioned in relative harmony for most of the duration of the Republic.
We're going to depart here, for a moment, to the state of Texas. The stereotypical Texan loves, among many other things, the concept of 'Murica - the idealized, romanticized nation of freedom, baseball, and mom's apple pie. One can imagine that, in a hypothetical stereotypical part of rural Texas, a politician must be for the idea of 'Murica - or, at least, not a critic of it - in order to get elected.
Things worked the same way in Roman politics after the peak of the Republic as a form of government (one could probably cite Cincinnatus as a turning point). There was a concept of the proper Roman, loyal and obedient to a fault, unflinching in the face of danger, willing to die for his country, family, people, and the idea of Rome. Anyone who wanted to be anyone in Rome had to hold to these ideals, or at the very least pretend to do so.
Octavian (Augustus), the first (and possibly best) Imperator of Rome, ruled over what can be considered the peak of Rome, bookending the turn of the millenium. His reign was a golden age for Roman arts, and featured economic prosperity for many. It was also Augustus's social policy to promote a return to "the old ways", the glorious Republican ideals we can compare to today's 'Murica.
So, while for the majority of its existence it was either a mechanic to raise the Senators at the cost of the people or a rubber stamp for the Imperator, the peak of Roman civilization depicts it as a noble and righteous body, the cornerstone of the Republic. It certainly didn't hurt that Rome was (and remains) the only well-known ancient republic, a form of government which the new United States chose to use.
Edit: clarifications