Hmmm. I read a book about guitar repair, and the author had a trick that might work: If the head of a screw is stripped or broken and doesn't have enough purchase for the screwdriver, he'd just cut another slot in it with a tiny Dremel tool bit. With a sewing machine you'd have to be careful about the filings getting into the gears, but it would give you a place to rest your screw driver.
Music gear is in a similar boat to sewing machines: they're precision equipment that often ends up in the hands of people who are all thumbs, and stripped screws happen
