I'm an AMA contest director, and feel that I support the AMA.
That being said, I'm going to share a brief anecdote. Take from it what you will.
26 Oct - 28 Oct 2018 was the AMA district fun fly. The local RC clubs asked the Las Vegas circle burners to be present. Right about the time of the ringmaster fly-a-thon (the first weekend of oct) the RC clubs notified us that there was an issue with cleaning up the dirt area that was supposed to be for CL use. We said well we will just fly beaters of the dirt area as is. Our club prez spoke with the district rep, and they advised against having control line flying there. Another week went by and the club prez spoke with the AMA rep again, and asked perhaps we can just do 2 demo flights. The AMA rep said it would be unreasonable to ask RC pilots who've traveled to wait to use the runway. He again advised that CL not be flown at all out there. During our meetings, the club decided, sounds like we're not welcome.
A gentleman from another state traveled out to the AMA district 10 fun fly. He brought out only a CL plane. He was told very specifically CL was not allowed to be flown at this fun fly. Our club didn't know this fellow, otherwise we would have warned him before he made such a long trip.