Not being able to login without a mouse/trackpad with the gnome3 gdm greeter has bothered me for quite some time, and usually I just complain silently and use the mouse. But today, I rejoice!
…because someone has figured it out!
Naysayers will say, of course it works, the username: field has focus, and you can just type your u/p and hit enter a couple times, and it’ll work. And if you enter things in just after it boots up, or just after you logout, that works. But if you let the screensaver kick in, you can dismiss that with Esc or space or enter, but the username: field will not have focus. And you can let your cat play with the keyboard at this point, because Gnome3 will ignore you completely.
So the solution is a magic incantation, which I scrawl here for my own benefit (and yours, if your search engine doesn’t lead you to the above link).
Use Ctrl-Alt-Tab to manually switch focus among the screen elements. I don’t know where this is documented.