1st: I don't think an ump should be asking to "show me the ball" when the ball is still live and there's still a play to be made. That's interfering with the flow of the game.
2nd: a really smart ball player will always be aware of the next play, i.e., the runner on 1st trying to go to 2nd, and also aware that he needs to show the umpire proof that he has control of the ball on the tag ... so a really good catcher would have popped up from that tag, stared down at 1st ready to throw, while showing the umpire the ball in his throwing hand in plain view. A good player doesn't wait for the ump to make his call ... he shows him the call he needs to make.
It's the same reason a good 1B-man will nod his head when he makes the catch and step briskly off the bag after the catch to demonstrate a sense of 'confidence' that he's made the play to the base ump. You help the ump make it look decisive.