The term is private hire, and that is the crux of the matter.
If a disabled person with some unusual requirement hires
someone to fix their central heating boiler or mow their lawn,
the law of contract applies. If the person being hired does not
have the means to attend to the unusual requirement, laws
about equal treatment do not oblige that person to re-equip
themselves so that they can provide whatever is necessary.
They simply point out they have not offered to accommodate
the unusual demand and they decline the contract.

The legal position is exactly the same with private hire drivers.
Remember: they are not taxis and they do not ply for hire.
They have a vehicle and they are not making any false claims
about the vehicle's capability. Someone wants to hire the vehicle
and driver. It turns out the vehicle is not suited to the customer's
needs. The contract is declined.

That is why TfL does not insist that private hire drivers have cars
which can take non-foldable wheelchairs. It is significant that TfL
demands that private hire drivers accept guide dogs which, of
course, fit into any car although many Asian drivers dislike having
dogs in their cars. The point here being that declining passengers
who are accompanied by guide dogs would amount to discrimination.