The scenario you bring up is a 2-party scenario between a client (the passenger) and a service provider (the driver) while Uber claims to be in a 3-party scenario where they are a software-based logistics service in-between the passenger and the driver. Their entire legal argument is resting on the claim that they are merely facilitating the scheduling and the payment of the ride.

Incidentally, if it were a 2-party scenario where you required a specific contractor to be available on demand, you would probably be their legal employer. A mandated, regular schedule is indeed one the main tests when distinguishing between an employee and a contractor. I have worked as a software contractor in the US for over almost 15 years and the employers, especially the larger ones with proper HR departments, have been careful to define my work in terms of skills and deliverables, not hours worked.