[tabs] Convert TabOrganizationTriggerObserverTest to BrowserTest

Per guidelines in BrowserWithTestWindowTest,
TabOrganizationTriggerObserverTest is being upgraded to a BrowserTest
because it gives us all the utility we need. This prevents us from
having to mock things the browser would do.

Bug: 384547121
Change-Id: I8f5b6f3510667fedf473829890c3350c21b23abd
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/6435046
Reviewed-by: Darryl James <dljames@chromium.org>
Commit-Queue: Sana Akbani <sanaakbani@google.com>
Cr-Commit-Position: refs/heads/main@{#1444472}