In response to the opening question:
The matter of לשיטתם is reference to an intellectual process. I suggest that the anti-vax movement is firmly rooted in emotion, not intellect. The decisions are made before examining empirical data, and then all research in used to provide circular reasoning, with correlational data used to infer causality, etc. This is not rational thinking. While I could not suggest that the anti-vaxxers lack intellect, it is plainly clear to me that they are not engaged at that level. Therefore, I conclude that they are doing so as an emotional reaction, which often ignores frank logic, fact, and truth.
I am not sure that the cholov stam matter is similarly emotional. So I do not think there is any logical connection.