We've been round this so many times, none of us are going to convince the others we're right or wrong.
As for my thoughts, yes, the individual practitioner makes a huge difference in one's capabilities. But there are arts out there more geared to towards actual self defense than others. What is studied will make a difference. More importantly, the schools training methodology will make a big difference as well.
On that front, often MMA gyms have a leg up. Most of their training comes out of drills that incorporate movement and multiple componants from other concerns. So, you'll be doing a mitt combo, and moving, and throwing a defensive pattern, and at the end you'll sprawl from a takedown. It's more integrated than many TMA's and "live".
Not to say that one has to be training in an MMA gym, and other arts that truely base training around a model like that can really put together some great things.
As for the limited weapons, yup, it's a bit concern for MMAers when you talk about sd. However, if you look at many of the things people tout: eye gouges, low punches and kicks, biting; most of these are easy and instincitve to pick up. If you get someone who is MMA trained, then chances are they can do some of those with minimal effort.
As for not culturally passing anything down, that's a different story and wasn't the original question. Even if you don't train MMA, there are plenty of training aspects that can be used by TMA practitoiners that would improve their game.