[lens-rearch][6/n] Add new searcbox controller

Adding a new class that will be responsible for all the Lens
search boxes. Currently this means the CSB in the overlay and the side
panel. This separates the search boxes dependency on the
LensOverlayController, meaning the side panel results could be open
without the overlay showing. Follow up CL will actually hook this up.

For more info, see http://shortn/_bc7W4TAbdS.

Bug: 413138792, 404941800
Change-Id: Ife5104bff8f375a41e985ba8b5b390cc8c7fa04a
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/6488678
Reviewed-by: Nihar Majmudar <niharm@google.com>
Commit-Queue: Duncan Mercer <mercerd@google.com>
Cr-Commit-Position: refs/heads/main@{#1452707}