[ASH] Reset app filter sheet when opened

With this CL, app filter sheet is run in a stateless fashion
i.e. every time the sheet is opened, it initializes the state (lastly
selected app) to the parameter passed to it. Only the caller
(HistoryManager) will the one that manages the current state.

In conjunction with the change, also ensures the HistoryManager keeps
the app selection state up-to-date when search mode is entered or
exited.

Bug: 331846135
Change-Id: Ife800aa49e744ecc1f6b784c314e2b8ac81ee48d
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/5479667
Reviewed-by: Zach Katz <katzz@google.com>
Reviewed-by: Theresa Sullivan <twellington@chromium.org>
Commit-Queue: Jinsuk Kim <jinsukkim@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1291692}