Edge Chromium opens already logged in to site

Well, perhaps this may help you somehow, read it carefully:

So, with the suggestions from kazu and a bit of effort of your side, you may get something.
You have to work on the Edge, with a fresh profile, take a snapshot of it, apply the change needed, take one more snapshot and diff them.

This may result in a certain capability which you can further apply.
Since Edge is Chromium based, some of the tips and trick suggested there should work, but perhaps in a different way.

Note that, I am not using Edge and I never will, so you have to figure out everything by yourself.