What should happen is allow drivers to get a doctor's note that marks their car as unstable for allergies (that way they can't make it up to discriminate). Also allow customers to mark themselves as needing to avoid animals or needing animal-friendly cars. You don't even have to show this information to either party: Uber can automatically choose the right car.
The law doesn't say every single driver has to pick up people that need service animals. It just says reasonable accommodations must be made.
[0] https://www.aafa.org/asthma-allergies-and-the-american-with-...
Of course, a simpler solution would be to have Uber choose an appropriate car that was willing to give rides to service animals.
I also think it should be fine to have a "no children" policy, we have all kinds of business we do not allow children in, and personally i would like to see more business adopt a no children policy.
We impose a lot of duties on people who are employed in the service of the general public. Humans are messy, unpredictable, inconsistent beings with unique needs so, yes, it is possible that two humans will be a mismatch for each other in the public sphere.
The law, and society in making that law, has generally decided that the right of a person with particular needs--such as a service animal or a carseat--outweigh the rights of someone to ply a trade in a customer-facing role. And, given the level of discrimination that people with those needs experience on a daily basis, I happen to think this is a fair trade-off. The person with the needs may not have a choice, or may have fewer choices, than the service provider.