Yes worry about it. Your dog is likely to end up getting shot by an angry hunter for running deer. During deer season keep your dog under your control at all times.
I don’t hunt so maybe some deer hunters will chime in and correct any misconceptions.
A deer nearly totalled my car a couple of years ago. I say let the dog have ‘em!
You’re absolutely right. The dog becomes a “Legal target” if it’s running deer.