One extra aspect worth mentioning about the 'arts' in the Martial Arts, is the issue of performance. This does have an aesthetic aspect, that means, what you look like will be judged by an 'appreciative community'. That is there is a value or 'beauty' that is given to your moves, that others assess. This could be a brutal effective aesthetic, it does not have to be flowery and pretty and dainty - that is no the point. My point is that there is a social basis to judgment.
If you look at youtube posts, watch how comments ... and stars are given to judge the performances of people doing their moves. The formal arts have this too. There is a culture of critique and judgment of fine art or performance based art - such as ballet. Sports have this aspect of performance in too, this is where demarcation of sports / arts blurs. Look at ice-skating for this. But even talk about football games are judged for their quality of performance and 'entertainment' values enter into this, as well.
Modern 'combatives' is a genre of martial arts that is not exempt from this if we focus on performance and its social basis. It is by default judged on criteria by 'experts' and this is no different from other 'arts' (both broad an narrow meanings given to that).
So what ~?
This is useful in knowing why we do what we do, when 'doing martial arts'. Where is motivation and what powers it. There is a performance aspect to this that feeds into our behavior, if you like it or not. We use it to judge ourselves and our identity - how we see ourselves. But is also a resource, involved in how others see us. And this affects our status, too. We are motivated to improve our performances in training.
I am arguing that in conjunction to functionality there is also performance, too in here, that some may want to deny but it is in here. Don't confuse aesthetic and performance with kata and finesse. See performance as valued activity that is judged subjectively by others. The boundaries of the performance is policed by a community too. Go watch on forums how opinion forms to verify if this or that move is effective / valued ; or if this or that performer is doing it right. This social activity could easily be confused to these 'X' Factor type programs. Ok - one is entertainment driven and the other is self-defence but the root is performance. It is social in its basis. You meet the criteria of positive feedback from these appreciative communities and that buzz and self-worth increases just the same whatever domain you are in.