How well would it work to ask the working LLM to rewrite the prompt to get the best results? Do the models understand enough about themselves to do that?

Claude has a /product-self-knowledge skill, and I am sure the others have something similar. So yes, it is possible if you work with care, as necessary with all things LLM related. There are hundreds if not thousands of skills on github that were created just this way.

That's kind of pointless, then, because what happens when Anthropic releases their next-gen model?

It's not like you aim to do it, you are just in a feedback loop improving results for the tool you are using. It is inherent in any prompt developed through iteration.