Refactor out code related to `search_starts_` in request

This CL additionally has a few behavioural changes:

- Marking a search as started CHECKs if it was started before.
- Marking a search as ended removes it from the `search_starts_` enum.

The second behavioural change will allow us to detect when all the
searches are finished more easily.

Bug: b:348067874
Change-Id: I6a6a21000fa782847af137f64ebe9f849927c5d2
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/5657832
Reviewed-by: Darren Shen <shend@chromium.org>
Commit-Queue: Michael Cui <mlcui@google.com>
Cr-Commit-Position: refs/heads/main@{#1320157}