Settings Split: Initialize dispatcher in shell

Bug: b/241965700
Change-Id: Ic7e4e8ea42a12658ac8a2f05ac6a5864dcbc860b
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/4328880


Reviewed-by: Scott Violet <sky@chromium.org>
Commit-Queue: David Padlipsky <dpad@google.com>
Cr-Commit-Position: refs/heads/main@{#1118891}