Yep. The new system is pretty damn good. The app is great. Here's a basic step through.
1. Press "taxi" icon
2. Press "Send cab"
3. Press "Now" (or enter a time)
4. It asks "at your currently location?" (uses your phones gps)
5. Press "Yes"
6. I think it asks for destination, but I don't remember the details.
7. Responds with "Cab in route. Approximate arrival time is 4 minutes 33 seconds"
8. A few minutes later your phone buzzes and a message shows, "Your cab has arrived".
It doesn't get any simpler than that. The taxi drivers love it (for the reasons you stated). The riders love it because it's faster and easier than the old phone system. The dispatchers hate it because they can no longer skim the drivers' fares. In NYC, I'm sure the cab companies skim from the dispatchers. The thing is, the cab companies probably see increased profits, except it will all be "on the books". The entire taxi infrastructure of NYC is rotten to the core (pun intended). A side note. The cab companies in NYC have hired lobbyists to get Washington, DC to implement a NYC medallion system. The tax drivers are fighting that tooth and nail.
My roommate could have gotten in on the ground floor of the company that makes this system, but he turned it down (he's still kicking himself over that).