[iOS] AuthenticationFlowInProfile: Use StartBatchOfPrimaryAccountChanges

...to indicate that an account switch is ongoing, which avoids UI
glitches (like the avatar temporarily switching to "signed out") on
account changes.
This logic is copied from AuthenticationFlow (from where it'll soon be
cleaned up, see https://crrev.com/c/6375304).

Bug: 375605482, 403183877
Change-Id: If5b72a19462544757a29a73f13986cf96665bbc4
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/6380494
Reviewed-by: Jérôme Lebel <jlebel@chromium.org>
Commit-Queue: Marc Treib <treib@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1435997}