[lens-rearch][3/n]Move side panel logic out of the overlay controller

Currently, the LensOverlayController is responsible for the majority of
the side panel machinery. This creates a tangled mess which makes it
hard to allow the side panel to be open without the overlay. This CL
moves the logic out of the LensOverlayController and into the
LensOverlaySidePanelCoordinator, with the goal being to completely
decouple side panel logic from the overlay, so eventually they can live
independently.

Next steps in a follow up CL are:
  1) Adding a state machine for the LensOverlaySidePanelCoordinator.
  2) Changing the name to LensSearchSidePanelCoordinator.
  3) Moving ownership to the LensSearchController.

Fixed: 405437199
Bug: 404941800
Change-Id: I9ebbb4e44e0e5d2e174e3cd30839982d9104f4be
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/6381719
Reviewed-by: Juan Mojica <juanmojica@google.com>
Commit-Queue: Duncan Mercer <mercerd@google.com>
Cr-Commit-Position: refs/heads/main@{#1448974}