I agree, you shouldn't be able to use Inspire Competence to do that normally. A lot of times oots doesn't perfectly follow the rules, so that's a possible explanation. Another is that Rich goofed. The reason I assumed from first seeing it (and I recognized right away "you can't do that") was that it was a joke. The NPC was misinterpreting her own class feature's rules.
Hmm. All skill checks are ability checks...