Your theory fails because regular Uber prices are almost half the price of a regular cab, which according to you is set on behalf of the public.

The benefit of Uber setting the price is that it makes the marketplace more reliable. And it's set at a level where both drivers and riders are happy. If Uber moves to a more dynamic pricing model, it makes it a lot more unpredictable and more complex. Maybe this will be the next step they take, but so far, I think everyone likes the predictability of the timeliness as well as the price.