Does being a "stable genius" mean that you clean up horse stalls very effectively?

Not at all. It means you know how to shovel horse shit.

If you have something Augean in mind, good grooming obliges me to inform you el bandito, that around these parts we holler for a marshal.

Firm, steady, smart guy. Ain’t got nothin’ to do with no horses, pard.

A stable genius would figure out how to have the stable clean itself.

A stable genius would remember that he’s run for president before unsuccessfully and not try to make out he’d succeeded on his first go.