[iOS] Make AuthenticationService's profile initialization more robust

...and record a histogram about it.

When a profile is loaded for the first time, AuthenticationService
performs some first-time initialization (in practice, this means
signing in if it's a managed profile), and then marks the profile as
"fully initialized".

Before this CL, it did all this really only once per profile.
After this CL, it goes through the logic on every profile load. This
has some benefits:
* It lets us verify that the profile's state is "correct" (e.g. a
  managed profile is indeed signed in), and record this in a
  histogram.
* In some cases, it lets us fix an inconsistent state (e.g. a
  managed profile that somehow got signed out can be signed in again).
* It's generally a bit more graceful - error states are recorded (and,
  where possible, fixed), instead of CHECK-failing.

Bug: 408380476
Change-Id: I089c4bc1e5344b88a5e6949eb6ebfb5efcbcddfb
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/6494524
Reviewed-by: Jérôme Lebel <jlebel@chromium.org>
Commit-Queue: Marc Treib <treib@chromium.org>
Reviewed-by: Menghan Yang <myuu@google.com>
Cr-Commit-Position: refs/heads/main@{#1456280}