“When a private company says take your content elsewhere, no one's First Amendment rights are violated.”

I think the concern is moreso regarding massive corporations like Facebook and Twitter working together in lockstep to remove people from their platform that they ideologically disagree with, thus preventing them from reaching their audience.

If Twitter wants to curate which people are allowed to post, then they are a publisher and should be regulated as such. That also means that they need to be held responsible for everything on their platform that they don’t remove.