My impression is that it's primarily the interface and reliability that makes Uber/Lyft better. Sure, short wait times are a nice bonus, but the primary problem with regular cabs isn't the speed so much as the unpredictability.
The problem is that I can't count on the voice-operated cab system. I have to call, and then wait outside at the pickup point, with a 50% chance of the cab actually getting there, at which point you call and get another 50% chance of the cab getting there within 30 minutes, at which point you repeat.
I have no idea if it's going to be 5 minutes or 120 minutes. I have to stay outside, too, because they won't call me when it's there.
If a cab took an hour but I could set it up on my phone while I was still in the bar, and it would alert me when it got there? That would be just fine with me.
Now, the local cab company could (and perhaps has, I have not checked lately) create an app that does just this; I don't really mind waiting half an hour if I had some assurance that the car would be there in half an hour.