Breathing exercises vs physical exercises (a debate... please join in if you exercise)?

I think the theory rests entirely on expanded lung capacity, if that is in fact a proven result. Controlled breathing is important in something like martial arts, but I don't know that it extends to something like running. I don't know if if it is true, or has been disproved, but at one time anyway, hyperventilating was used by runners, which I suppose falls under controlled breathing, but is the opposite of the calm breathing a martial artist will try to maintain when gassed.

I think endurance training is the only thing that improves it. One could run a mile, not do it again for months, focusing only on breathing technique, and when they go to run again, there time will most likely not be any better, nor the run any easier.

Endurance is body function specific in most, if not all cases. Running helps a martial artist gain stamina, but it doesn't mean one could go from kicking or punching 500 times in a row to 1000, just from the additional overall health/stamina.

Knowing how to breathe properly can have some benefits, and to a degree you can increase your lung capacity (although, from what I understand, not by much), but what it doesn't do is increase the lactic acid threshold of your muscles, or wire your neurology for continuing to do the same motions. At the end of the day, if you're preparing for some sort of physical activity, you need to do that physical activity, with some general and activity-specific conditioning exercises mixed in. Controlling your breathing will keep you relaxed and help you to expend energy at the correct time, but you still need to work out your muscles.

Lung capacity is only part of the equation. Deep breathing can help stretch the tissues of the rib cage and strengthen the diaphragm. It will not, on its own, give you more endurance.

The endurance comes from breathing correctly for the activity you are doing AND doing that activity for progressively longer periods of time.