Caligula’s horse wasn’t made a Senator.
The Roman system was all about separation of powers, and that was rooted in the oligarchic nature of the “Republic”. The Senators weren’t elected, so they established or altered various other popular bodies to meet some expediency. The Emperor had dictator-for-life powers, but the Senate controlled legislation.
The Emperor Claudius revived the pre-imperial office of Censor in order to remove unfit Senators from office. That the office had lapsed indicates just how out of control the senatorial oligarchy had become. They were, in Voltaire’s phrase, an assembly of despots.
The Senate didn’t always appoint the Emperors, sometimes it merely put its imprimatur on a fait accompli, such as when crazy or otherwise ineffective Emperors were deposed / murdered, for example when the Praetorian Guard (which was established by the Senate) snuffed Caligula and the famous and probably false “found me hiding behind the curtain” selection of Claudius occurred. But the authority to appoint Emperors lay with the Senate.
Claudius was one of the more conscientious Emperors. If the Empire had figured out an orderly system of succession (for that matter, if the so-called Republic had), the great duration of it might have been even longer. Had the Empire — which began over a century before Julius Caesar was born — developed an elective system of government, it might still be around today.
I think any sane person would rather live under the rule of two elected Consuls and the Senate. Although election was a mixed blessing. One Consul decided to try to “force” a military victory against Hannibal Barca in order to ensure an electoral victory. It was a bad mistake. Military campaigns shouldn't be run with an election timetable/ political concerns in mind.