Make Lacros tests ready for async update of last active browser.

This cl continues the effort for making the tests which relies
on BrowserList::GetLastActive() returning the expected browser
after UI operation more robust, and ready when we change
last active browser update to be completely asynchronous
for Lacros.

Bug: 325634285
Change-Id: Ia4ec55daeca7bddb282e2c618e21b5162677fdf0
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/5336309


Reviewed-by: Lei Zhang <thestig@chromium.org>
Commit-Queue: Jenny Zhang <jennyz@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1267373}