The debate is whether they're doing MMA (a completely separate thing) or a combination of things. It's a reasonable question that doesn't have a definitive answer. It's like asking whether I teach Shojin-ryu, or mainline NGA with some Judo/FMA/boxing/BJJ. I can argue both sides of that, because it doesn't really matter which we call it, so long as we all know what it is.
In the case of MMA, I think it gets more vague, because there are some places that do teach what I consider MMA (as a style), and others that specifically teach the component styles, plus how to use them together for MMA competition. So I'd argue sometimes it's a style, and sometimes it's just the competition the gym trains folks for.