While that's correct (only providers of fixed-route services like shuttle runs must be wheelchair accessible), there are a number of ADA requirements for private transport services[1]:
Additionally, private entities providing
taxi service cannot discriminate against
individuals with disabilities by refusing
to provide service to individuals with
disabilities who can use taxi vehicles
and/or use service animals, refusing to
assist with the stowing of mobility
devices, or charging higher fares or
fees for carrying individuals with
disabilities and their equipment than are
charged to other persons.
So you can't refuse someone with a service animal, even if you don't want service animals in your vehicle. And you can't refuse someone with a mobility device that
can be stowed in your vehicle (such as a collapsible wheelchair), just because you don't want the hassle of the added time or inconvenience of helping stow away their device.
[1] https://adata.org/factsheet/ADA-accessible-transportation