I would say that "trust science!" tee-shirts aren't really about Feyerabend's epistemological anarchy. You could frame it that way, but that would be philosophy-washing (to coin a term).
Against Method points out that there's no rigorous method for pursuing the difference between truth and non-truth, but at least accepts that there is something like "truth" out there and that it's better than falsehoods.
When people put on "trust science!" shirts, they're not arguing about whether we should continue working on homeopathy. They're saying that people are promoting deliberate falsehoods, things that they know or should know are wrong, because they are indifferent to the difference between truth and lies.
Homeopaths at least want to be right. There are surely some charlatans, but at least some of the kooky nutbags selling homeopathic remedies believe in it. Against Method makes it impossible to argue that they cannot possibly ever succeed, if they keep revising their hypothesis, even though we "know" that they won't.
The opponents of those who "trust science", by contrast, just don't care if they're right. They care only that they've won some kind of victory in a culture war. They don't apply epistemic anarchy to all sciences, only ones that they view as ideological opponents.
That isn't the only threat to science, to be sure. Science and scientists make genuine mistakes. But again, that's not really important to those who are rejecting science for cultural reasons, not philosophical ones. That challenge makes dealing with the others even harder.