You're right, it
could go the other way. Note that Uber drivers protested the recent Uber price drop but Uber only drops the price to compete with other companies and does not care what happens to the drivers who are just a resource to them.
So individual drivers may well increase their rates if they're left free to do so.
But the biggest discriminator between getting a lot of fares and not getting fares will be price, not newspapers or free water and I suspect that the majority of the drivers would drop their prices in order to compete with other Uber drivers, rather than with other Taxi companies.
This then takes away control from Uber over their revenue stream and would quickly erode the market. In countries where I've been where cabs were completely de-regulated cab rides were ridiculously cheap because of this effect.