So how does someone like Musk - who owns most/all of X - take advantage of that "wealth"? Can he borrow against it? Can he sell "shares" and get some income from it? How are the values determined?

Well, X is actually now a division of SpaceX - so that's not a great example. But say Neuralink instead.

The question isn't whether the company is capable of being valued - passive voice. The issue is who can value the company. Musk has access to all of Neuralink's business records - so Musk can value the company. The company can value itself. If the company (or Musk) wants to borrow money, they can open up their books to the lender - and the lender can then reach an assessment of the valuation, presumably covering the cost of conducting that evaluation from the fees it expects to earn from the loan. If there's to be a loan on Neuralink, the lender will do the due diligence - and Musk (or the company) will make available what they need to, almost certainly with a confidentiality agreement so that the company's entire books don't end up on the interwebs.

But that's different than having the government have an idea of what the company is worth. With real property taxation, the government typically already has the really big pieces of information it needs to conduct an evaluation. It's all already public record. And even then, there's tens of thousands of employees at property appraiser offices across the country to do that.