Uber does provide training that says drivers must accept service dogs. However, if there are issues, they are uncommon enough and the likelihood of a driver getting sued is low enough that a driver may just drive away.
My gut tells me that drivers should be employees, or limits should be placed on using single-person independent contractors to provide a service like this. It's not going to solve every problem and the traditional "taxi" mentality is going to remain, but I remember the early days of Uber when drivers could not reject too many rides and stay on the platform, and it did seem to solve some of the management problems with drivers. Heck, maybe the answer is to contract the work out to car management companies with real HR departments like the airlines do with regional jet flying, so that Uber can insist on their high quality product and performance metrics and drivers can be employees.