Search backend: tidy up eventHandler

Created by: camdencheek

Just a couple of small cleanups from previous PRs.

1. remove cancel because it's not actually needed, and depending on it would actually be incorrect because we want Done() to finish things not stop things (the parent context can stop things)
2. reorganize the struct fields so it's more clear which things are protected by the mutex.

Stacked on #37582

Test plan

Unit tests. Just a semantics-preserving cleanup.