Seems to be two things here not just one:
Why is it not possible for Air Transport to collect frieght (primarily helicopters) from the many airfields that are AT capable (i.e. Wattisham)?
Here we seem to be talking about collection for deployment.
Is it realised how much engineering effort is required to 'fix' helicopters once they have flown as frieght in order to then fly them to where they live and vice versa.
Here we seem to be talking about aircraft returned from deployment.
This process involves amazing amounts of man hours and engineering effort for what appears little benefit.
The aircraft have to be prepared somewhere. Presumably easier to do this at Wattisham rather than fly them to Brize and prepare them there. Similarly on return it would be easier to recover them at Wattisham.
The nub of the question thus seems to be, 'why do we have to transport our helicopters (by road) when they could be delivered to Wattisham direct.' In other words very little to do with engineering effort, or am I missing a trick?
The point about crash cat for a fully loaded C17 departing Wattisham has been made. There is also the additional 'out of service' time of the additional turn-round at Wattisham and the recovery leg to Brize, not to mention the possiblity of the crew being out of duty time.
Making the same argument for troop moved however I would argue that delivering 200 squaddies direct to home would go someway to improve morale rather than a convoy of buses from Brize.