Considering that sports martial artists generally train to spar/compete for timed rounds and Defense focused artists generally train for that 3-5 second intense engagement, yes I would say that the idea that martial arts APPLICATION is going to demand more explosive/short burst power generation than aerobic (running, swimming, cycling for extended periods of time - think Tour de France). But, you do have to have a certain level of aerobic fitness to be able to repeatedly create explosive bursts (rounds in matches, all out powerstriking in combinations/bursts in self defense - including running away).
I would say though that for the sake of SAFE conditioning development, a student should have some aerobic elements in training (usually independent of a class) BECAUSE the general sign of athletic aerobic fitness is that you return to/near your resting heart rate within 2 minutes of finishing some intense activity (push ups for time, sprints, sparring rounds, focus mit drills). Of course this doesn't mean that you are tapping the pads for 2 minutes and don't get your heart pounding. I am talking about all out, 100% effort exersions.
Should an instructor try to build in aerobic training during class? Maybe, but it might not be enough in frequency, duration or intensity for everyone in the class. Aerobic exercise is really something that should be encouraged, modelled in class by the instructors (I love jumping rope), and outsourced to fitness experts that can help students create a workout perscription that is right for them to avoid injury/overtraining.
There are some great on line sources for aerobic fitness evaluations that students can use to figure out if they need more aerobic training to maximize their endurance foundation which will improve their explosive potential (combined with muscle 'strength base' assessments).