I suspect that you have never actually used a phone-dispatch car service in NYC. You have no visibility into their ETA. Call back and ask... the answer is always the same "5 minutes".
I've used it tons of times. I grew up around here. You're right, you don't have much visibility into the ETA. I THOUGHT that was an advantage of Uber, but it's not -- I've sometimes called Uber and they blatantly lie, bait-and-switching about ETAs (eg, 3 min before I click, then it becomes 12min). Also, dealing with the ETA is easy. Wait 6 minutes, if they don't show up, call a different car service. The problem will take car of itself.
The "best" part about Uber is they penalize you for waiting too long to cancel your Uber so if the guy takes three times as long as promised and you cancel then you are charged the cancelation fee.