The military don't want anymore civilians on their airfields than they absolutely have to and Cobham despite their training regime and ex military staff are still civilians. They also do practice escorts into landing and using a civilian airport with crossing traffic etc adds to the realism should the need arise for a full scale follow and escort or force down. MME is ideal because of the lack of scheduled traffic to cause delays and also the private traffic adding hazards to factor in.
When MME starts picking up again perhaps they will do the overshoots at somewhere like HUY. ( Just a little joke there. Don't bite. )