Namaste CG. I"ll give you my take as well, for what it's worth. Both Yogananda and Maharishi watered down Hinduism a great deal to appeal to what they perceived (rightly or wrongly) as a very Christian audience. I read that Maharishi was really surprised to learn of the 'free love' and drug culture of that era. I believe he assumed incorrectly that such things wouldn't be there, having been raised in India. The 'Christian' audience wasn't all that Christian, but in India there was no 'nothing'. and there still isn't much.
Breathing (pranayama) and hatha yoga are later stages in the classical yoga sutras. What some swamis omitted were the first two stages, the yamas and niyamas. So often they start at or near the top, rather than at the bottom. So people aren't all that stable, and they hear of all this advanced stuff like chakras, advanced pranayama, and more without having their feet on the ground, so to speak. It can be harmful, or ego will take hold. In more traditional Hinduism we start at the bottom of the path, and get stable, than accept God's help as we need it. You don't give beginners at piano advanced lessons. The music won't be very great if you do.
Yes it's far more exciting, and sounds wonderful at the top. But without living a balanced and dharmic lifestyle, it isn't really going to get you much in any permanent way.
This is all Hinduism, albeit a very liberal form of it. Most new-age groups totally omit the word, even though the roots of the teaching are in Hinduism. The term has a negative connotation, and if you attach it to the teachings, westerners would all drop out of the classes. The anti-Hindu crowd has had an impact that way.
If you ask followers of these universalist swamis like Sri Sri (disciple of maharishi) Sai Baba, Sadhguru, or Yogananda, they'll declare to you that their not Hindu, all the while singing Hindu bhajans, worshipping Hindu icons, and more.
I agree with Marcion that the 'New Age' term is an umbrella term for a wide assortment of things, some of them being very Hinduisn, while others way off that mark.