It’s not the main driving factor
Sure, that is trivially true.
I do think there is still merit to understanding those who are "not ordinary", who don't know they have a fear of needles and will just rationalise and rationalise until the world ends. While I'm sympathetic to your evocation of state sponsored misinfo, I don't think that's it either, at least not in a deep way since vaccine hesitancy has existed since the dawn of inoculations. In fact vaccine hesitancy might also be understood as a historical remnant from a time when vaccinations had much more common side effects. Variolation for example had a significant death rate. Seventy years ago, when the science of drug delivery wasn't really a thing, side effects were much more pronounced. Basically: medical procedures that everyone has to go through are an easy target to activate the "lizard brain" with, be it intentional or not.