birch: Add metrics for birch bar, part 6

See go/birch-metrics. This CL adds Ash.Birch.TotalLatencyPostLogin.

Bug: b:330188422
Test: added to ash_unittests
Change-Id: I1c07ecc6a9ef3fd3cd41555e20e5f978b45fe6b9
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/5416015


Reviewed-by: Matthew Mourgos <mmourgos@chromium.org>
Reviewed-by: Michele Fan <michelefan@chromium.org>
Commit-Queue: James Cook <jamescook@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1282046}