pay ED pay to serving staff to drive for them
That's certainly not going to happen. With manning always tight and legal driving and rest hours on the drivers, they don't have very much slack in the system to support non-military tasks.
As has been said elsewhere - there are some precise definitions of what military MT can be used for. A lot of what an ATC Sqn might want to do is going to be outside those definitions. Hence they have their own vehicles.
By the way; there are over 1000 ATC Sqns across the UK. That is a lot to support with a dwindling RAF.