As I said to another person on this thread: if scientists let their political views override their pursuit of truth, the public will (rightly) lose faith in science.
So when you tell them to "trust the science" -- be it vaccinations, climate change or something else -- they have no reason to trust that science.
People don't have faith in inanimate things like science. They have faith in their leaders, who then lead the way in what to believe.
If those leaders believe in the integrity of the scientific institutions, their flock will follow. If they're anti-vax, their flock will follow. If they believe in some medical quackery, their flock will follow. If they believe in eugenics, their flock will follow. It's happened before.
What was fringe yesterday can become mainstream today, with the right leaders.
So when you tell them to "trust the science" -- be it vaccinations, climate change or something else -- they have no reason to trust that science.