Here’s something interesting to chew on. Cardio respiratory endurance is sport (modal) specific. You take someone that is an elite (or even above average) endurance athlete in one sport (modality) and place them into a different sport and their dominance or competitive edge largely disappears. Take Lance Armstrong at his peak all doped up on EPO and drop him into the NYC Marathon and his formidable cardio respiratory endurance is above average but not elite and he was certainly not a threat to any of the top runners. When he ran many people speculated that since he was arguably the best endurance athlete in the world, he would crush the NYC Marathon. His best time was a 2:46 which is respectable but not dominant.
This is why triathletes basically train all the time because they are trying to increase their capacity in each modality. If cardio worked like most people think, then a triathlete could basically train one sport to excess and expect it carry over to the other two sports. So if we know that the cardio from biking doesn’t really help our running and the cardio from running doesn’t really help our biking, then why do we believe doing either of those things will help our jiujitsu?