Move more AutomationInternalCustomBindings logic to ui/accessibility

Refactors methods that do not depend on extensions to
AutomationAXTreeManagerOwner, which is inherited by
AutomationInternalCustomBindings. More refactoring can be
done in a follow-up to move logic from methods which partially
depend on extensions, and to port logic from within bindings
lambdas.

AX-Relnotes: N/A
Bug: 1357889
Change-Id: Ib30973add021fbdc54dee9d7919e74c07d340d82
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/3872827
Reviewed-by: David Tseng <dtseng@chromium.org>
Commit-Queue: Katie Dektar <katie@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1046603}