Musk is almost certainly a little nutty, but also a lot calculated. Plus he has access to plenty of advisors, consultants, and experts in their fields if he has questions.
One thing his Twitter shenanigans have done is wake a lot of people up to who already owns/controls Twitter. All the hand-wringing about Musk might be justified, but why no concern (until now) about the very shady cast of characters who already own it? Does anyone seriously think that a multi-billionaire Saudi Prince gives a shit about freedom of speech? How about major hedge fund managers?
Musk may actually care about free speech to some extent since it serves his interests at the moment, so even if we hate the guy we ought to acknowledge that he's shining a light on things that probably needed to be revealed to a general public that usually never thinks about who actually owns and controls the digital infrastructure of our lives.
He also put the Board in a bind with his offer since their duty is to shareholders and if they are fulfilling that duty they would be compelled to take it. If I owned significant stock I'd be peeved that the Board was not helping me maximize my gains, which is contrary to my interests and therefore actionable (possibly gaining class-action status).
The fact that Vanguard immediately increased its position (it now owns more than Musk) suggests that Musk isn't the only person who thinks the platform is valuable and worth owning.
Here's another possibility I've seen few consider: Massive AI and machine learning projects like the kind Musk has been into need input data for "learning." The more the better. Twitter not only has an inconceivable amount of data, it also uses some interesting algorithms and buying it would give Musk complete access to all of it, which is probably where Twitter's true long-term value lies.