Uber should have to deal with this systemically because it is shitty to just leave it up to the drivers.
Please note that in your example here, it is also shitty for the rider who is expecting to be driven somewhere safely.
A lot of people in this thread are arguing that the driver should be forced to suck it up and I think that's shitty for both the driver's and rider's safety.
It's not uncommon for small towns to have one Uber driver going at certain times. Well that driver can't be the only driver around if they aren't going to take service animals. So Uber would have to tell them that they can't drive unless they are willing to take service animals or there is another driver around that is willing to.