extensions: Update DeveloperPrivateEventRouterShared

Since this class was created a couple of its dependencies have been
ported to desktop Android (ExtensionManagement, ExtensionAllowlist).
Migrate code that uses them from DeveloperPrivateEventRouterDesktop
to DeveloperPrivateEventRouterShared.

There may be a couple more opportunities to move code (perhaps the
toolbar action support?) but I figure we can wait on those until
the features are further along.

Pure refactor, no functional change.

Bug: 392777363
Change-Id: I4e762534ffa7a97a8b4d21ae517aa5c428f5966a
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/6473217
Reviewed-by: Devlin Cronin <rdevlin.cronin@chromium.org>
Commit-Queue: James Cook <jamescook@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1449558}