But, again as you note, all of that was the first-among-equals set dressing of the Principate where the Republic's centering institutions were given lip service, and it would be hundreds of years before the outright Dominate of Diocletian would be generally accepted by Rome--and getting there required the transcendental shitshow that was the Roman third century, itself worth perturbing a lot of electrons.
In conclusions, land of contrasts, etcetera etcetera.
IIRC the purges and the rumors about Caligula mostly started after a serious illness in 37AD, where he might have just literally gone around the bend? But I'm definitely not ruling out the possibility that Caligula was played up after his death, though. If you look back at pre-Republican mythology (it ain't quite history), the seven kings of Rome are all caricatures. Caligula could definitely just be The Crazy One.
In any case, Rome put up with some real wacky characters as emperor so long as they didn't bankrupt the empire.
What I find really interesting is the murder of political opponents and how it affects legacy. From where I sit, pretty much every single emperor killed senators, perhaps excluding only Caesar himself, but only a few got saddled with the blood thirsty tyrant image. Claudius killed quite a few senators, some without good justification, yet his reputation is largely “good emperor, misunderstood”, while Tiberius gets off the hook largely because his successor was so much worse. It’s all pretty fascinating to me.
Given the reaction of the people after Caesar's assassination, it was not a popular move by the Senate, regardless of the distrust the people had toward kingship in general. I understand the point you're making but personally find it to be a bit of a stretch.
But my point is more subtle. Caesar was a arguably a populist, and was both ambitious and wildly popular with the common people of Rome. But there were a few incidents where Caesar was offered to be crowned king, and the people of Rome hated it. It would be a few more generations before the idea of being ruled by a guy in purple whose dad had done the same would be okay to Romans.