Aerobic training does have benefits for combat athletes (actually ALL athletes). It allows your body to deal with a greater work volume in your training, therefore you can do more sport specific anaerobic work (pad rounds, sparring etc) and recover better. Check out Tom Kurz Science of Sports Training or check out his articles at Stadion.com.
I really don’t buy into this modern trend of ONLY doing training in the anaerobic zone. Maybe for amateurs who only have a few hours a week to train and need to prioritize, but for professional fighters who train all day, at least a couple of aerobic sessions should be done a week in my opinion.
If i train my fighters I let them do long distance runs in the preparation period before the competition period just ones a week to give them a good base condition.
Sugar Jackson (2 times european champion) does every day a run of 20km, i don’t see the point in that but he’s good so it doesn’t hurt him and definitely doesn’t slow him down !
I get great benefit out of running 3 miles 2 days per week. I notice that if I run more than that I have a tough time getting in the power work i want in the weight room and doing all the skill work I want on the mat. However, I’ve read that Fedor runs 5 miles a day, every day. And, ya know, he’s (said in Rogan voice) “got amazing, world-class, black-belt level, cardio”