Admin toolbar search: Improve automated tests

Problem/Motivation

This task is a follow-up to #3554658: Admin Toolbar Search: Remove dependency on Admin Toolbar, where the dependencies of Admin toolbar search were updated.

The dependencies removed were added to the existing tests where required so they could stay compatible.
But no specific test was added for the module itself without other dependencies.

Steps to reproduce

Run phpunit tests for the 'admin_toolbar_search' module.

Proposed resolution

If possible, the check on the three menu local tasks (tabs) on the admin settings form page, should be moved out of the existing test class:
https://git.drupalcode.org/project/admin_toolbar/-/blob/3.6.2/admin_tool...

to another Functional test class already requiring these modules, so the dependencies on 'admin_toolbar' and 'admin_toolbar_tools' could be removed, leaving the standalone 'admin_toolbar_search' for the test:
https://git.drupalcode.org/project/admin_toolbar/-/blob/3.6.2/admin_tool...

Additionally, a new FunctionalJavascript test class should probably be created as well, to cover the JS code of the module with only the core toolbar module enabled: Test submitting a few search queries, for example.
 
In general, try removing the dependencies wherever possible in the tests.

Remaining tasks

User interface changes

API changes

Data model changes