@cori no, not related to the issue you linked. The issue only happens if you have a functions folder in your Glitch project and specifically a package.json file in that folder (different from the standard package.json file that can be found at the root of any Glitch project).
If I rename / remove this firebase-functions-specific package.json file, the “Deploy to Firebase” runs smoothly. If I keep it, it throws the “HTTP Error: 404, Method not found.” error (which I suppose is a generic error not related with what is really happening).
I suppose that when we click on the “Deploy to Firebase” button, behind the scene you are simply doing a “firebase deploy” via the Firebase CLI but at that moment, it reads the functions folder and try to do something that doesn’t work with the package.json file (eg: install npm dependencies).