[tabs] Convert ExistingBaseSubMenuModelTest to BrowserTest

Per guidelines in BrowserWithTestWindowTest, ExistingBaseSubMenuModelTest 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: I2dc339f1b6c8977ac1c0c418a8798f9d53ebd756
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/6463323
Auto-Submit: Sana Akbani <sanaakbani@google.com>
Commit-Queue: Darryl James <dljames@chromium.org>
Commit-Queue: Sana Akbani <sanaakbani@google.com>
Reviewed-by: Darryl James <dljames@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1447948}