There are reasonably low barriers to entry in this market, for the most part. If Uber treating its drivers as contractors rather than employees impacts Uber drivers in a negative enough way that they would prefer to work elsewhere, there is likely ample space for a competitor which offers a better deal for its drivers.
Drivers have little-to-no incentive to stay with Uber if they can get a better deal elsewhere (based on what I've seen, many already also work for Lyft), and users have little-to-no reason to stick with one on-demand ride app if there's another app that has more availability/drivers.
If cost-of-ride is the biggest competitive differentiator, the price of the ride will stay the same and drivers will be able to choose whether they want to work for a company which treats them as an employee but takes a bigger cut, or a company which takes a smaller cut but treats them as a contractor.