MMA might also ruffle a few feathers to other MA instructors who teach technique, but can't use that technique in sparring/fighting situations to their best advantage, and end up losing in a MMA setting. That's a fault with the practitioner, not the art.
I think that this is where the largest amount of friction developes between MMA and TMA. I think to a great extent, you have an oranges to tangerines comparison.
If you train in a TMA, you will (hopefully) learn techniques that will enable you to survive a violent attack against a mugger/kidnapper/rapist/general bad guy. You get into good enough shape to execute these techniques without pulling anything and without falling on your face
Some take it a step further and engage in tournaments. These are often point tournaments and do not allow a full range of grappling, or are limited to grappling only.
Then there are guys and gals who train for MMA competition. MMA competition has a broader rule set than a Karate or Judo tournament, and often the competitors train more like modern athletes than the typical martial artist does.
Needless to say, if your world is the first or second category, unless you are simply an awesome fighter who can easily adapt and who trains very, very hard, you'll be at a severe disadvantage when stepping into an MMA competition. You'll be unused to the rule set, probably not conditioned in the same way as your competition, and you will likely not be familiar with many techniques outside of those you know in your core art.
Needless to say, when such a person competes and loses, it adds fuel to the MMA beats TMA arguement. I don't believe that either beats either; there is a lot of cross over, but an MMA tournament is very different than what one trains for in a TMA dojo. Could a TMA practitioner become competative in MMA? Sure. But learn the rule set, familiarize yourself with the techniques that you'll face, and train to counter them. That and be prepared to eat some humble pie while you bring yourself up to speed..... just as you would when you start a brand new art and are a beginner. MMA is a very different animal than the dojo setting.
Kind of like the difference between the Baja and a police chase. Both involve cars, but the Baja will demand a very different skill set than a police chase. The Baja will require being able to drive in differing terrain, off of pavement, and you will be constantly performing at your best level to win. A police chase is comparatively short and requires the officer to apprehend his opponent, not just out drive them. The officer must do so without endangering civilians and does not have the opportunity to choose the circumstances of the chase and may be getting shot at.
Could a police officer take his cruiser to the baja and win? No. The car wouldn't survive. Does that make the Baja driver inherently better than the police officer? No. They each have different goals and objectives, and each is best at what they do.
Anyway, hows that for a long winded analogy?
Daniel