[Uno-D] Clear Avatar Button Signin Message on Signout

On signin, the avatar button expands and shows: "Making Chrome Yours".
If sign out happens while the message is shown, the message should be
cleared.

This can notably happen when Management screen is declined, which
signs the user out.

Fixed: 446170918
Change-Id: Ie9b77255339d8c599818624cd77fc3a371ee42eb
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/7002980
Reviewed-by: Amelie Schneider <amelies@google.com>
Commit-Queue: Ryan Sultanem <rsult@google.com>
Cr-Commit-Position: refs/heads/main@{#1523515}