Your argument has an element of logic but is fundamentally flawed. I'm sure you wouldn't want the Army owning and operating all RAF vehicle fleets while the RAF own and operate all Army and RN helicopters. Or would you?
Why not? Unless it is a refueller or tug or runway caravan or other aircraft-specific support vehicle then why not? Most of the RAF's fleet is 'white fleet' anyway and hired in via LEX or other such hire companies. In fact, thinking about it, why not give all airfield/aircraft-specific vehicles to the Army (RCT - Radio Cabs and Taxis) as well and we will ask them to provide when needed - simple arrangement would be 'no support vehicles then no flying for Pongos' - simples!
The REME should be looking after all military vehicles - it's what they're best at. The RAF Techies should be looking after all military aircraft and the RN Techies should be looking after all military vessels - likewise, it's what they're best at. It's so frighteningly easy and would make the management of today's small numbers of skill types easier by pooling them by type of vehicle/vessel to each Service.
LJ