I think your point downthread about how this could impact the truly "on-demand" nature of the services is apt. As you said, the value prop is you get a guaranteed car in about 15 minutes (and often it is much less), whereas the old systems required calling, being routed to a nearby car service and having someone dispatched or a on-the-road driver routed to you.
In heavily urban areas, the average user might not see much of a difference (with few exceptions for weather, getting a cab in Manhattan pre and post Uber was basically the same and I've frequently canceled an Uber when I was able to hail a cab faster), but if you expand that just a smudge, say Brooklyn, which doesn't have yellow cabs unless they happen to be on the way back from a drop-off -- (and don't get me started on the uselessness of the green cabs) and as such was one of Uber's first really strong markets (I signed up for Uber the month it launched in NYC, back when it was livery only -- and used it almost exclusively in Brooklyn). The subsidized price (even though it has only decreased for drivers) has allowed cars to circle neighborhoods in Brooklyn -- but if that goes away and prices increase, drivers aren't necessarily going to sit or drive around each of those neighborhoods and will instead migrate to the most dense parts of town.
This is even worse for cities with historically little or no cab ecosystems (Atlanta -- and Atlanta is at an even worse advantage because of the terrible stage of its public transit, which only exists in the city proper), where Uber/Lyft effectively have become the cab system. I certainly don't think that market will leave those cities, but it costs rise (which seems likely), that will impact demand and will also in turn, impact how long it takes someone to get a ride.
(That said, I do not expect us to return to a pre-Uber in Atlanta world where it would cost $100 to get a cab from the airport to Dunwoody)
I do think the combination of increased demand (or at least awareness/ease of access to call a car) and logistical improvements (to me, this has always been Uber's greatest strength) will prevent things from turning into NYC in the 90s in most larger cities. But it is an open question in smaller cities that didn't have that infrastructure to begin with or are much more spread out.