I doubt Twitter classifies moderated content into political buckets as this would create massive liability, but has any neutral third party studied what kind of content is removed and classified it into "left" vs "right"?

I'd like to see the system that could accurately perform such classification at scale, if such a thing is even possible, since no one seems to know exactly what "left" and "right" mean.

For example, "liberal" and "left" are often conflated even though hardcore leftists tend to have a healthy disdain for liberals (in the modern, colloquial sense).

I think mainstream now pushes both, which, a bit, depends on what you'd call "mainstream". If "mainstream" is "media watched by millions that thematizes the public discourse", then Fox News and Netflix are both mainstream, and their range of ideas being pushed span wide. I don't see that the mainstream prefers either side, on a worldwide stage.

"More extreme people are kicked off each platform successively until they end up places like 4chan."

I think extreme expressions are moderated chiefly because it turns a place toxic, and because the owner wouldn't like to be legally liable for things like hate speech or doxxing.