If it has 15 employees or more, then the law assumes that it's reasonable for the company to provide service (e.g. by reassigning a non-allergic employee to work with a customer who has a service dog) except where it would demonstrably create an undue hardship.
Uber did try what you said as a defense, and they lost, so it seem courts think that because Uber is a huge company, with much more than 15 people, it's on them to find a way to be compliant with ADA.
Weather or not drivers are subcontractors or not doesn't even come to play.
Because lets face it. If they didn't thats such a huge hole, that in a lot of cases corporations would be able to structure themselves in a way that they could avoid that.
Some drivers may lie, but if you require even a simple doctor's note I think that would deter a lot of people from bothering, given the rarity of encountering a service dog request.
Sure, which clearly matters of Uber drivers are employees, but less so if they are independent service providers.