Agreed. Application is how the training is used. If the training is used to perform in competition, that is what you're training for. If the training is used so that you get really, really, really good at training, the system itself becomes the application.
I think as people move beyond proficiency in an art and into the higher levels of expertise, it's pretty easy to see the dysfunction. Where you have styles that are seldom applied outside of the style, the evaluation and analysis of the art turn inward. You start to see folks 'finding" things in the style and looking for deeper meaning. You start to see the style discovering new applications and it's amazing that everything you need is there.
The dialogue around the training changes, too. You start to hear discussion about spirituality, and deeper meanings to all of the movements. This tends to lead people to be more insular and less open to cross training.
Now, none of the above is a problem, if one's goals and objectives are aligned to the training. If you're looking for spirituality and this resonates with you (whatever "this" might be), great. It only becomes an issue if you think you're learning something you are not actually learning.
And to bring this back around to this thread, if there is conflict between what you're learning and what you think you're learning, some amount of mental static is created. How much depends on you. If you're training in a style as described above, and start to cross-train in another style, the issue isn't that you're training two styles. It's that you're seeing alternatives perspectives on things you might previously have taken at face value. That cognitive dissonance might never be fully resolved, which would keep you from excelling in either style.
This cognitive dissonance is less likely to occur in styles where the goals are consistent with the application, and application is not just encouraged but facilitated. This application could be professional or not.