I thought that was because Tonks trained under Moody?
"Sign up for the Auror preparation program in your sixth year," said Susan. "It's the next best thing. Oh, and if a famous Auror (assumed it was referring to Moody?) offers to oversee your summer internship, just ignore anyone who warns you that he's a terrible influence or that you're almost certainly going to die."
This is a world where there are potions that can regrow bones in a single night. I think it wouldn't matter that much to Harry if he did lose his finger.