It seems hit and miss.
Best guess is Motability are happy to leave in the hands of the garage once the order is approved.
Unless there is a special need for motability to get invlved, i.e. any adaptations, or maybe if the Motability customer is vulnerable, they are happy to leave the process alone.
Again, usually it down to cost. Motability will work with the minimum of staff levels to provide a service.
But as we have seen with COVID, they must have increased the staffing levels to deal with the refund process for insurance and Good Condition Bonus.