If someone tells you that they're an MMA fighter, you're going to ask what arts they trained in. If their answer is "MMA," then a real life Abbott and Costello situation is about to happen.
As time goes on, an increasing number of MMA competitors are entering the sport having trained just MMA. They don't necessarily have any prior martial arts background. They don't necessarily take separate classes in BJJ, Muay Thai, etc. They just train MMA.
A little while back I did some research to determine the primary training background of every UFC champion ever. For about 20% of them, the answer was just "MMA". Good luck telling a UFC champion that he's not an MMA fighter because he hasn't studied BJJ, boxing, wrestling, etc in separate classes.
I get the desire to have language be logical, so that we can just look at the individual words in a phrase and derive the literal meaning, But that's not really how language works. (If it did, then we'd have to look at the entire phrase - mixed
martial arts. "Martial" means related to war or the military. So "mixed martial arts" would mean something like combined operations where you get your infantry, tanks, and artillery to work together.)
Language is defined by its users. For a majority of the general public, as well as martial artists and competitive fighters, the term "mixed martial arts" aka MMA refers to the modern combat sport which incorporates unarmed striking and submission grappling, including fighting standing up and on the ground and to the training methods which have been developed in conjunction with that sport. It doesn't refer to arts like JKD, Wado Ryu, Kajukenbo, Danzan Ryu, etc even though those were all explicitly created by combining elements of pre-existing arts. It also doesn't refer to your cousin Bob who trains both Judo and Kali and is figuring out how to combine them in a fight.
You might like those MMA practitioners who just train MMA as a unified system rather than taking classes in separate arts to come up with a new name for what they do, but unless something happens in the cultural zeitgeist to make that happen organically. you can't force it.