Yes, of course, I have installed and activated both GSX and GSX2, see attached screenshot.
Something is very strange, what you are describing basically cannot happen because, if there was a problem with the software running, you won't get a Trial, GSX wouldn't even start. If you see the Trial menu, it means GSX is working perfectly fine, but the *base* product is not activated.
However, you are on Windows 11, and nobody at FSDT use it yet, so we don't know if there might be some difference there, which usually are related to the .NET libraries. Usually, installing the FSDT Universal Installer should fix any of these issues, possibly after a restart.