> I assume demand in major cities doesn't really change that much so that the schedule is announced and updated very rarely (similar to public transportation).
This is way way way off. Looking just at my experience in San Francisco: weather patterns (in detail, across different parts of the city), public transit issues, tourism, events, fluctuations in the city's population and travel across various holidays/festivals/three-day weekends, changes in price of Uber, changes in price of alternatives (i.e. Lyft drops prices)...and the interplay of all these variables (e.g. public transit happens to have diminished capacity along a critical line for a given event vs having diminished capacity on an unrelated line).
It's not market fundamentalism to acknowledge that what the market is good at is this sort of massively decentralized, million-variable price optimization[1]. This is precisely the reason why direct price controls on consumer products tend not to work: because it's damn near impossible for any entity (even the freaking US gov't) to accurately gauge the million variables that go into setting supply and demand. The idea that Uber could do this without being way, way, way off almost all the time is a hilariously bad idea.
As someone alluded to below (and I alluded to above multiple times): The product Uber is selling is predictable-availability rides. The product taxis are selling is predictable-price rides. It's nonsense to say that reducing the predictability of Uber's availability would be "less stressful" or easier on its passengers.
[1] Note that it optimizes for allocative efficiency. Much of what market fundamentalism gets wrong is ignoring that allocative efficiency is not always what our goal is. That's not relevant in this case though, since we're explicitly talking about how a centralized, top-down rate schedule is a laughably terrible idea.