They are talking utter cr@p
Motability have no say in it.
If the AP drops before you collect the car all the dealer has to do is cancel the Motability order (not the factory order for the car) and then resubmit it. Takes 5 minutes, doesn’t cost the dealer anything, they don’t lose any money.
All that changes is the amount you pay to Motability via the dealer when you collect the car.
Please excuse spelling/typos. Apart from being a clot it turns out I had one on my cerebellum that's now causing various problems!