I thought it was just the venom that reacted badly with Arthur's stitches. Or that there wasn't even a reaction, but it just didn't work. That would require the bandages to be changed, and I think Molly would be nearly as upset. I think that if Arthur had merely fallen off a broom and split his head open, stitches would have worked. Of course, I have no real basis for this (and will eventually get around to a major post on wizards, Muggles, and health, but not soon), but it's my theory, anyway.