With the current internet age, I think many even in the West would have known the etymology of ‘Yoga’. Those who had joined ‘yoga’ classes in the West and anywhere else would also been informed of the etymological origin of the word ‘yoga’ i.e.
The Sanskrit noun योग yoga is derived from the root yuj “to attach, join, harness, yoke”.[23] The word yoga is cognate with English “yoke”.[24]
The spiritual sense of the word yoga first arises in Epic Sanskrit, in the second half of the 1st millennium BCE, and is associated with the philosophical system presented in the Yoga Sutras of Patanjali, with the chief aim of “uniting” the human spirit with the Divine.
en.wikipedia.org/wiki/Yoga
I am surprised you have given such slim attention to the Art of Breathing, i.e. pranayama.
I recommend you read Yogi Ramachakra’s The Science of Breath where ‘prana’ is critical to the physical, the psychic and the spiritual.
archive.org/details/in.ernet.dli.2015.47927
The Yogi practices exercises by which he attains control of his body, and is enabled to send to any organ or part an increased flow of vital force or “prana,” thereby strengthening and invigorating the part or organ.
As I had stated within Hinduism, Buddhism and other, there a wide range of topics and practices of breathing from kindergarten to PhD levels.
In this work we cannot go deeply into the philosophy
of the East regarding spiritual development, because this
subject would require volumes to cover it, and then again
the subject is too abstruse to interest the average reader.
There are also other reasons, well known to occultists, why
this knowledge should not be spread broadcast at this time.
Rest assured, dear student, that when the time comes for
you to take the next step, the way will be opened out before
you.
Page 69
I believe from the Hinduism perspective, in the ultimate union of the self [atman] with Brahman, this need to be supported by the most optimal breathing technique and the alignment of the prana.