Exclude 'media.tag' from search

[WebShapedBiz]

Is your feature request related to a problem? Please describe.
Media tag schema is something that sticks out from the rest of the content and none of my clients (half a dozen of them, so not that many) want to use them. I know how to remove media.tag from the structure builder and I know how to remove it from the global and local "create new" menu, but I seem to not be able to hide it from the search menu.
I know about the experimental schema filter __experimental_omnisearch_visibility: false but that's not something that I can use since the media.tag schema comes from this plugin.

Describe the solution you'd like
A possibility to hide media.tag from the Studio elements, not including the Media plugin itself.