and would not include it in the main repo
Tests that verify behavior at run time belong elsewhere
The test blobs belong in whatever repository they're used.
It's comically dumb to think that a repository won't include tests. So binary blobs like this absolutely do belong in the repository.
I have had to un-teach dumb things that people learn from Windows.
A menu item to run a GUI program as root it is indeed a rather absurd scenario. It suggests that you want to violate the admin/user barrier which is intended to be difficult to surpass except in certain circumstances.
There can be a lot of things under the hood that are necessary to run a GUI program as root depending on whether you're using X11 or Wayland or something more esoteric. It's doable though.
But instead of doing that, why not just learn how to use the command line? Every administrative task can be done via the command line, but not every administrative task has a GUI counterpart. So you're going to need to learn to use the command line sooner or later.