I think it's all about balance. Of course doing only long distance running as an exercise is not good. But neither is being in a average couch potato shape.
You take someone, who's been training all his life in different sports, and then picks up freediving. His heart is already very strong and he is in general good fitness. He may want to concentrate more on diving than running.
Then take someone who's not been very active in his prior life and wants to take up freediving. I would recommend that they get in shape too. Someone with no prior, or very little, background in serious training would in my opinnion benefit from regular, fairly long distance running, simply for a foundation of fitness for him to build on. Then, as he's fairly fit, he can slowly move towards other forms of exercise.
I think the kind of apnea swimming that for example Natalia Molchanova does, is a fairly different thing than running half marathons. If I understood correctly, she swims laps holding her breath every other 25m or half a lap or something like that, and does this 6 times a week for a couple of hours. In my opinnion this is very opposite to pure aerobic training. Instead of conditioning her self to draw up and burn oxygen as efficiently as possible, she is conditioning her body to move and do work as much as possible, with a more or less constant oxygen debt. Very different indeed...
Anyway, what is absolutely clear and I have no doubt about, is that you need to have a longish break from jogging to hit your best apnea performance. The effect is not so dramatic for static, but it's very dramatic for dynamic/cw.
It makes obvious sense that too much aerobic shape is not good. What it in essence means is good shape at "drawing oxygen from your lungs and burning it". I for example have been training way too much aerobic lately, and it's evident in the horrible 7:00/100m STA/DYN ratio I'm at currently (which I would guess has lots to do with slow/fast fiber ratio). However, I'm confident that once I stop running, I will make leaps of improvement within weeks.
That's how (to my knowledge) a lot of competitive divers plan their season. They indeed do loads of cardio, but during that time, they are well aware that their diving capability is reduced. But they take comfort in knowing, that switching training regimes weeks or even months before the competition will change that and they reap the benefits then. It's all about timing your "peak", just like any other sport. It's about planning a whole year of training in favor of hitting a week or two of optimal performance. It may well include half a year of serious aerobic training. Or then something completely different.
Personally I find that starting pure apnea training too soon will result in peaking too early, which means at the competition I'll be sloping down already, which really takes the edge off. Kind of like drinking. 1.0 promilles is much more fun when it's going up, that when it's coming down. But you're just as drunk on both

But you cannot just maintain that 1.0 and be "happy happy joy joy" for hours. You must either drink more to become even more merry (eventually resulting in disaster), or call it a day and go to rest.
Finally, I think Paul's referring to "Mr. Fattah's thesis" is simply saying that his training is what most freedivers would call extreme and experimental. It does not necesserily apply to most divers at different levels as such. Even if it is fundamentally right, it applies different to divers in different levels of progression and general fitness. And who can honestly say that during the years there hasn't been some "out there" theories from Eric? I'm not saying this in a negative meaning, but it's just that some times he goes very far in the search for the ultimate apneist. Further than most average divers would be willing to follow.