OK, so after nearly 2 years of fighting this problem, I think I finally have figured it out (though Ford continues to say it's not a problem and that they can't diagnose it, or it's because something was done to modify the stereo... pick your set of excuses). So, enough crabbing and on with what's going on:
In it's infinite wisdom (insert sarcasm), Ford has left the power outlets to be on even when the car is turned off. Now this means that ANY item you leave plugged in is going to drain the battery (even a simple USB power adapter has a continual drain). What causes the MUSTANG and non-working radio to start is a draw on the battery beyond a certain level (some here mentioned 4 hours of a GPS, I've had it after several days parked with just a USB power adapter plugged in).
Now the fun begins because, best as I can figure it, there is a reset routine that requires a certain voltage to be maintained for a certain time and then suddenly everything starts working. First, I'd suggest the attempts above at disconnecting the battery for 10-15 min (negative first), and then be sure fuses 39 and 3 are good. If this doesn't work (like it didn't for me), on to the next step.
Sometimes just driving at highway speeds for 60-120 min will fix it. More reliably I've found that giving the battery a trickle charge over night, then driving at highway speeds for 60-120 min is the most reliable (unfortunately just the charging isn't enough to do the reset).
Yes this is at total PITA and such a design fail in my opinion, but it's better than the $600+ alternative of replacing the stereo (yes the dealer will try to sell you a new one, saying that's the problem). Next step for me is to change the wiring to the outlet I use for USB power so that it turns off with the ignition (there's a reason this worked so well starting from Henry Ford's first car).
OK, a bit of griping but hopefully some good info that all those dealing with this can use to avoid big expenses in the future. Best of luck!