[Uno-D] Adapt AvatarToolbarButton Browser test part #4

Follow up on series of CLs: http://crrev.com/c/6974678

This part takes care of the interaction with the Management badge.

Enabling the Phase 2 feature in most of those tests + adaptation.

Fixed: 444617947
Change-Id: Ibfbfc65f89f306f293dcb26d3ae0de226a9d09f7
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/6979450
Reviewed-by: Ernest Nguyen Hung <ernn@google.com>
Commit-Queue: Ryan Sultanem <rsult@google.com>
Cr-Commit-Position: refs/heads/main@{#1520460}