If you had a QB who could also punt sometimes that would make for some interesting options/fakeouts.
It's been done before!
There's a lot of psychology to going for it on 4th down or faking it.
3rd down stop! Everyone is excited! Defense jogging off the field, high 5's helmets on the posts, gatorade...
All the fans of the team on defense are feeling relief: we stopped them! We're getting the ball.
What does the team who just made the 3rd down stop want more than anything? A simple punt that isn't muffed or fumbled.