But a new Twitter would pop up within a month.

It would be called something else that would be easily to remember like Tikker or Woobly.

A new Twitter cannot pop up in a month. A site with all of twitters functionality could. But the value is not in the site, it's in the adoption.

Given the significance to some politicians I can imagine that they would intervene if it came to that.

Musk has to take care to not overplay his hand.

"It's a private company" has been the excuse for the abusive censorship. Are you saying if someone has right wing leanings on free speech that politicians will now change their tune and threaten it?

Not at all (under the assumption that a possible Twitter under Elon Musk would continue to adhere to local laws).

But trying to shut down a platform which is considered (by some) "critical infrastructure" might find you in a position where regulations restrict what you can do.

It's not a certainty but a possibility that this might happen (regardless of who is in power at the time.) Many politicians of different leanings use Twitter for communication.

Yes! ;-))