I think you're mostly wrong about how yoga is practiced in the United States, where it is almost never a spiritual practice. I practiced yoga for many years and several different places with many different people and it never had any spiritual element to it at all.

Historically, Yoga has a spiritual component, and I'm sure it does in India. But in the United States, it (mostly) just doesn't. Most of my yoga instructors have known nothing about spiritual components of Yoga and have never so much as mentioned them.