I'd say the simplest answer is that Kishi deemed it too OP once he created it. He made it possible for all jonin to use the body flicker, and that instantly makes them much more powerful in terms of rushing an opponent. Later in the manga, he concentrated on shinobi power and delegated speed feats to very specific shinobi. These shinobi (like Minato, Sasuke and Ay) built their potential off speed alone.
If everyone in part 2 was able to use body flicker as lucratively as they did in part one, speed feats like Minato's, Sasuke's and Ay's would drop considerably in potency. Bottom line? Body flicker was generally omitted from the baseline ninja capability in part 2. Instead, Kishi purposely created new speed feats outside this technique in order to make more characters unique and give them strength outside of powerful techniques.