On the PC it turns out that if you press home+L or home+R button you can change the "mode" of the directions, L, and R.
For instance, while by default, the Mayflash is set to "Mode 1":
L and R is set to "s0 positive" and "s1 positive" (I assume these are analog) and the d-pad is set to "POV hat <direction>" (or something like that).
If you press home+R, then it changes to "Mode 2":
L is set to "Button 4" and R is "Button 5", just like all the other buttons (ABXY). The d-pad is set to buttons as well, i.e. "Button 10" for up, "Button 11" for down, etc.
If you press it again, it goes to "Mode 3":
L and R are the same as in Mode 2, but the d-pad... is kind of weird. It only registers Up and Right for me, and I think it's done via analog signals.
Press it again, and it cycles back to Mode 1. Long story short, if you press home+R or home+L (which just cycles in the other direction of home+R), you can set it so that every input is digital (by getting it to Mode 2).
Of course, I say all this having done it only with a Hori stick, and only on the PC. Throw a TvC stick or PS3 into the equation, and I can't say for sure how things would change.
Edit: This post is in response to TwoThirtyEight's comment that L and R wouldn't work because they're analog.