[clangd] Add index server request logging

- Add verbose logging of payloads
- Add public logging of request summaries
- fix non-logging of messages in request scopes (oops!)
- add test for public/non-public logging, extending pipeline_helper a bit.

We've accumulated quite a lot of duplication in the request handlers by now.
I should factor that out, but not in this patch...

Differential Revision: https://reviews.llvm.org/D90654