LSP syntax highlighting is a new-ish thing, and apparently isn't
supported by Eglot.
True, but there is a pending patch by Akib Azmain Turja that adds this
very feature to Eglot. Akib hopes to start the copyright process soon.
https://github.com/joaotavora/eglot/pull/839
Do you expect it to have advantages compared to using TreeSitter?
I'm not very familiar with TreeSitter, but I think TreeSitter would for
all practical cases provide a faster/simpler solution to font-locking.
(TreeSitter uses an already defined grammar, whereas an LSP server could
potentially take into account dynamic information. I've reimplemented
the hangman game as an LSP server, and there the color of a word depends
on its location within the file. The color of the hanged man could also
depend on the number of remaining guesses.)