I think you'll find that if you get stopped by a police officer, they will insist you do the same. Some people have an ID holder, but many people have ID's visible in their wallet. By making you take out your id, they avoid the situation where someone hands over their wallet to show the id and then later claims, rightly or wrongly, that the TSA agent stole money from their wallet. Also, it avoids any situation where it might be seen that a passenger was offering a bribe to the gate agent.

By the way, what the hell is Rent a Schmuck?