...for a fender stay mount like that:

1. even if you do go off center and hit some of the threading on one side of the hole, it's usually not enough to matter much. The machine screw that holds that fender stay in place doesn't carry much load with a fender.

2. if you are mounting something that will carry a load, like a rack, and you're worried about the damaged threads, it's not a big deal to drill out the stay eye a size larger and re-tap it to fit a larger machine screw.

But yeah, I usually just assume that if a machine screw is in there tight enough that turning it snapped off the head, drilling is my best bet for removal. In the smaller bit sizes, those bits that are coated with miracle space age titanium nitride don't cost that much more, and seem to last longer in use for me.