Thank you for the clarification.
I don’t know what Rav Kook held (although you can hardly call someone who ate meat a vegetarian), but either way, I question the motives of the frum vegetarians. Regardless of whether it’s connected to a political movement, it’s an “ism”, and I remain skeptical that their approach was derived by delving into the sugya. I am much more inclined to believe that they decided on vegetarianism the same way Mathew Scully did, and then looked for support from Torah authorities.