I suspect the problem is that at some of the "largest" airports there are no longer any clubs, or if there is a club, there is only one. With one club you use the club, without any, well its the FBO handling. I dare say there are some places with a club but where the airport authority "prevents" the club acting as handling agent - Leeds Bradford possibly comes to mind, and maybe Southampton (I think the club at Southampton is long gone, but there again I could be wrong). Certainly at Leeds Bradford you get firmly steered to the FBO handlers (and very nice they are) whereas in fact there is a perfectly good looking flying club almost next door who I guess dont charge their members anything like the FBO for parking and landing.
Is it anti competitive - presumably it is within the remitt of a privately owned airport to decide who is authorised to "handle" private flights and if they chose not to give that authority to a private club so far as non based aircraft are concerned have they acted beyond there rights? After all I am guessing the airport authorities owns the tarmac on which the aircraft are parked and grants the lease to any FBOs or clubs?
That said I like the approach Southampton takes - they have set aside some spaces for GA on the apron run and operated by the airport - if the spaces are free you dont need to use the handling agents - it is still not cheap mind you. It is a shame a few more of the very "largest" airports havent followed their lead.