[clangd] Use a separate RunningTask flag instead of leaving a broken request on top of the queue

Summary:
This helps us prevent races when scheduler (or any other thread) tries
to read a request while it's still running.

Reviewers: sammccall

Subscribers: ilya-biryukov, javed.absar, MaskRay, jkorous, arphaman, usaxena95, cfe-commits

Tags: #clang

Differential Revision: https://reviews.llvm.org/D75927