Make PickerAppListControllerDelegate a singleton.

PickerAppListControllerDelegate is only used as a "dummy" object
passed to the search engine code. It has no state. Make it a
singleton so we don't have to create multiple copies of it when
we convert PickerClientImpl into a KeyedService.

Bug: b:365741912
Change-Id: I097768fbb64b518336234cca7d290cac4dee16b9
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/5869183
Reviewed-by: Michael Cui <mlcui@google.com>
Commit-Queue: Darren Shen <shend@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1362205}