[clangd] Destroy context before resetting CurrentReq

Summary:
Our tests stash callbacks into request context and rely on it being
invoked before threads going idle.

Reviewers: sammccall

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

Tags: #clang

Differential Revision: https://reviews.llvm.org/D77671