I can't believe someone has to make a thread asking this kind of common sense stuff. The OP was there, we weren't. The OP knows if it was really a mistake, we don't. It does kind of sound like the vet was doing a procedure just to do a procedure, but that's just me. There's ointment that you put on wounds to keep them from getting infected.... if it really was just a flesh wound. Which again, the OP knows better than we do.
Whether or not I took my dog back would depend on how it happened. Mistakes happen, and if it was just a mistake and something that left Pixie's head quickly (as the OP describes it), then it sounds about as bad as accidentally taking too much when you clip the claws. It's a mistake, it happens, no harm, no foul.
But we've also had very bad experience with a groom who didn't seem to like dogs much anymore, and the groom was even recommended by the breeder. When we told the breeder, she called the parent of another relative, who confirmed things have gone bad there and he wouldn't be taking his dog back. So then she called the groom and blessed her out, and the woman hung up on her...and then she just called back and continued.
The groom really screwed up there, because while our breeder doesn't have a huge kennel and it's been ages since she's bred a litter, she's very active not just in the Dachshund club, but in the Finnish Kennel Club in general. She's at all the shows, volunteering, helping out, and if there's a groom mistreating dogs, she will tell everyone, and she did.
I wonder if they went out of business. I hope so.
If that doesn't sound like your groom, then maybe it was just an accident. Only the OP knows for sure. But the troll posts from a few of the regular troll types sure were helpful.