[Refactor] Refactor ManageSyncSettings.java to be more readable.

This change breaks down the if/else for Account Settings setup into
different helper methods, the overall logic does not change.

Change-Id: I78e2c85e1406dadcef98d86ee971dbd60dc532c8
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/6607354
Reviewed-by: Marc Treib <treib@chromium.org>
Reviewed-by: Boris Sazonov <bsazonov@chromium.org>
Commit-Queue: Luchen Peng <luchenpeng@google.com>
Cr-Commit-Position: refs/heads/main@{#1469183}