[Settings] Isolate dependency injector from SettingsActivity

SettingsActivity has multiple functionalities, and one of them is to
inject dependencies to its child fragments. This patch extracts the
functionality to an isolated class to make the upcoming refactoring of
SettingsActivity easier.

Bug: b/356743945
Change-Id: I1599aa46f86144a85811c4ef396e63abff06c09f
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/5815330
Reviewed-by: Sirisha Kavuluru <skavuluru@google.com>
Commit-Queue: Shuhei Takahashi <nya@chromium.org>
Reviewed-by: Theresa Sullivan <twellington@chromium.org>
Code-Coverage: findit-for-me@appspot.gserviceaccount.com <findit-for-me@appspot.gserviceaccount.com>
Cr-Commit-Position: refs/heads/main@{#1349109}