Thanks for the reference. I did some further research on my own as well to uncover a lot of history with this question. It was very informative. What I gathered from that and the other references I found I can try to briefly summarize.
There are Hindus who decry the "gutting" of the religious aspects of yoga by Westerners. They would typically represent a traditionalist perspective. It would much like a Catholic being upset by some other culture co-opting genuflections for some other popular, capitalistic purpose.
They are one voice weighing in on what the West has done with yoga. And that argument makes the case that what you see in your school, is NOT what the Hindus who decry the exploitation of what to them is a religious practice, for secular, capitalistic purposes.
That supports what the judge in these cases has ruled, that is is not in the form you see, a religious practice. If it were, these Hindus would be happy about it. But they are not happy about it. To these Hindus, it's not really yoga because it's not a religious practice.
But the deeper and more salient thing that I discovered is in this other well-informed article I found. What is the actual history of yoga? Did it actually begin as a religious practice, or was it something that religions adopted that began as something independent from them?
My gut feeling had been telling me that was the case, as I study and practice Taijiquan. The techniques it discovered with utilizing what is comparable to yoga practices, are the sorts of things that make you optimally balanced, mentally and physically, which results in a deeper sense of connection with reality, what I call the "spiritual" nature of life. These are things that were discovered, and then brought into the practices.
Same thing with yoga being brought into religion. But I should point out it was not just HInduism, but Buddhism, and others as well. So it did not originate in Hinduism.
Here's a good article that outlays why the misconception that it 4000 years old, is likely in error. He gets into the why of that in his article, but I'll highlight the salient points you should consider:
A more sober reading of the historical record suggests that the types of spiritual exercises we now categorize together as yoga, including methods of breath control, mental concentration, and seated postures, developed in the first millennium BCE in South Asia.
The development of these exercises was not the exclusive “intellectual property” of any single sect or religion.
Rather, it was a project shared among diverse groups that we may retrospectively label “Hindu,” “Buddhist,” and “Jain,” along with other sects, such as the “Fatalists” (Ajivikas), whose teachings have largely been lost. Thus we find, in the second half of the first millennium BCE, the first unambiguous references to “yoga” and meditation (dhyana) in certain late sections of the Vedas, along with the early texts of the Buddhists and Jains.19
As the influential historian Johannes Bronkhorst usefully reminds us, “The spiritual discipline yoga does not belong to any philosophical system, but may, or may not, get connected with a variety of philosophies, depending on the circumstances.”20 Neither Patanjali, nor the Buddhists, nor the Pasupatas suggested that yoga belonged only to one group.
Yoga in classical India was like open-source software. It was distributed freely and modified by different authors, all competing to come up with the best version for liberation.
Source
It's the same sort of thing with the martial arts. They are at its core, physical exercises. Secondarly, they may, or may not be, adopted to fit into spiritual practice. Many athletes take advantage of this for better fitness and health and energy. Again, the actual source of these things, is the human body itself. That makes it transportable, even into the completely non-religious secularized version of it you see in your classroom.
All that said however, I do understand how those who are unaware of these things, might see them fearfully as they come from another culture than their own. If they see life through the lens that other religions are devil worshippers, then anything associate with them, are viewed as satanically influenced. This is of course not supported by what you see above, but I do recognize that some think that way. It's difficult to overcome fear with reason. It's a primal response sort of emotion.
