Although I don't think that taking a formal training program is the only way to develop the necessary skill to do something.
No, formalized training is not the only way to learn something and in some instances it may not even be the best way. And as you've pointed out, many of the pioneers of the sport had little to no formalized training (Cousteau, Exley, et. al). They learned by trying things and pushing the boundaries of their knowledge and experience and gradually building up a reference base for what the limits were and eventually others learned from them.
On the flip side, some people that have formalized training, still aren't qualified to do some things. Training does not equal competency. This wide disparity in competency is what fuels a lot of these debates.
Diver A may take the formalized training, have 150 dives at that level and still be a complete Charlie Foxtrot in the water.
Diver B may have learned from a mentor, done 20 of those dives and be much more competent in the water.
So of those divers, who is more qualified to do them? A with their formal training or B with their competence?
The debate rages on...