This is touching on the whole obvious problem with no controls on speech: cancel culture is free speech since it's just me asking other people not to freely associate by freely speaking that I don't want to freely associate with them. Then you enter the whole "tolerance of intolerance" debate and so on.

This whole thing is a question of degrees. I think "common carrier" was a useful concept: loss of control traded for immunity.

Wouldn't the analog counterpart be the postal system? They have no right to poke in your private communications.

Edit: That said postal systems are generally owned by the government. So I suppose ultimately (if nothing else) it would be up to their T&C. But I don't see how they could reasonably justify monitoring the traffic. It's not like it's a PR issue for them what users share through their tubes. I would imagine it'd make sense to have them regulated similarly as the postal system (ensuring private communications).

No, it'd be FedEx, since the USPS is the government.