It would be called something else that would be easily to remember like Tikker or Woobly.
Musk has to take care to not overplay his hand.
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.
I would argue that the drift toward extremism has much more to do with the advertising revenue business model.
Content that enrages, engages. More user engagement = more revenue.
Growth in social media means attracting more users and/or increasing engagement of existing users. Civil and reasonable discourse is bad for engagement numbers, outrage and extremism do much better.
Public companies and their leadership face much more pressure to show profit and growth every 3 months. This pushes them toward short-term thinking and easy ways to juice their numbers. For social media the easiest way to do this is the promote and encourage content that incites outrage, since this drives engagement, which in turn drives ad $$$