Refactor search start members in request to map

This actually uses a std::array as suggested by dcheng@'s Slack comment
[1].

This will help us know how many searches are started (i.e. the number of
non-null entries in `search_starts_`. However, GIF searches are
technically not started until after burn-in.

This is also technically a behavioural change - the query time metrics
now "end" at right before the search request calls `current_callback_`.
Previously, the query time metrics "ended" as soon as possible after the
response from the search source arrives, before preprocessing. This
mainly affects CrOS search results, which truncate the search results
before calling `current_callback_`.

[1]: https://chromium.slack.com/archives/CGGKQ3KA9/p1695749305687779

Bug: b:348067874
Change-Id: I6a6a210068966b40ddd8fc5aae2033112b162f42
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/5645091
Commit-Queue: Michael Cui <mlcui@google.com>
Reviewed-by: Darren Shen <shend@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1318894}

• ash/picker/search/picker_search_request.cc[diff]
• ash/picker/search/picker_search_request.h[diff]