You must not have Hunted for deer or rabbits before, I've been hunting both for a lot of years and it's rare to see either animal freeze on you when frightened, 99% of the time their first reaction is to flee!
So I must've imagined all those deer and rabbits I used to bring home? Thanks for clearing that up for me!
A rabbit freezes for a second before it takes off. Most deer too. That's when you take your shot. I've hunted rabbit with a bow at times. I've also hunted plenty of deer with both rifle and bow and usually get close enough for a clean shot. It's called "woodcraft" or stalking. You should try it sometime instead of blundering around in the woods and making as much racket as you can.
And if you're a poacher, you "shine" a deer at night to make them freeze.