I often read about breathless state brought about during deep meditation. It intuitively appeals to the mind that slowing the breath is associated with tranquility and longevity.
My question is what about aerobic exercise? It is very much necessary to avoid diabetes and heart problems and one breathes very fast during the exercise. I wonder what the Yogis' take on this is? In Hatha Yoga Pradipika it is mentioned that the Yogi should not exert himself. Makes no sens from a fitness perspective.
You’re right: aerobic exercise is an important factor in health. In fact, Paramhansa Yogananda taught that one should “perform some sort of exercise every day until perspiration breaks out over your whole body.” (See his book, How to Achieve Glowing Health and Vitality.)
As for the Pradipika, I suspect that it is merely warning against over-exertion, which will diminish the benefits of the practice. Exertion certainly is required in some asanas, but it should be done in moderate doses, Over-exertion will elevate pulse and breath, and inhibit energy awareness and control, such that one misses out on the higher — and much more important — aspects of Hatha practice, especially the serene mind, inner awareness, and uplifted consciousness.
If one wishes to use Hatha Yoga for spiritual purposes, which was its original intention, it is better to get one’s aerobic activity through walking, running, swimming, etc.