First of all, you should know that with respect to being my worst mentee ever you have lots of competition and are in no danger of earning that distinction! In fact most if not all of my mentees have experiences one or more crises not unlike yours; it's part of the process imposed by a graduate writing program; so as for your struggle being unique or your being ill-equipped for the program, the answer there too is "No."
As for "the quality that distinguishes real writers from fake writers," you are quite right to suspect that such a quality exists--though I doubt very much that you've identified it correctly. The quality is called persistence. That is the ONLY quality that, in my experience, draws that distinction: not talent, not intelligence, not level of education or life experience. Persistence. And that is being tested in you here, now. Will you pass?
Setbacks, disappointments, self-doubts are part of not only an MFA program, but the writer's lot. The goal isn't to eradicate them, but to NOT let them defeat you (persistence). And since persistence is the one quality you really need to worry about, I have only one piece of advice for you in this crisis: persist!
Also some stories are just, to not mince words, a pain in the ass to write and/or we are not ready to write them, yet. Sometimes it's best to turn to other projects. Persistence is crucial; but we need to choose our battles, too. As long as you persist in SOMETHING then you aren't giving up.