I think it's probably yoga's being a product of Indian culture, not in spite of, that counts against it. Indians and Indian culture are very much scorned in RVF and manosphere circles, as being the complete antithesis of not just neo-masculinity in particular, but Western moral and cultural superiority in general.
When you think about it, so too far quite a fair few offshoots of India at odds with the red pill and manhood -- pacifism, compassion, vegetarianism, Buddhism. I can see how an RVFer would thumb his nose at the idea of doing your fellow man no harm, as an inherently feminine trait, the
raison d'être of mankind being to forcefully seek out and conquering the ground of other men whilst defending your patch in turn. Similarly, a steak and two veg is praised as the most masculine meal one can consume, a testament to the strength and forcefulness of man and his mastery of lesser fauna.
I imagine Gandhi couldn't count on too many fans among the 'sphere.
As for how this relates to yoga -- wouldn't it be fair to say that its soft and nimble nature would speak to what its mental and spiritual side would be? The idea of harmonising your mind, soul, body, with your surroundings point to an effete belief system, easily domitable belief system.
In contrast to weightlifting, which can be seen as a more masculine form of physical conditioning, as it involves the honing of ones anatomical faculties and mastery of strength, for the purposes of control over his environment. This also sharpens his martial state and stature.
Maybe I'm reading too much into it, or have swallowed too much of the Manosphere Kool-aid, but I can't help but see a justification for the feminine stigma it enjoys.
I can definitely see the benefits that you've explained, and reckon that it augment and broaden my workout routine and capabilties....but maybe I see doing it as a slight against my manhood?