That's a good point; humans are not perfect "anti-induction machines". That said, each person who'd presented this argument to me had spent a lot of mental effort during his or her life to embrace and perfect this worldview. In the same way as a rationalist would train himself to use Bayesian reasoning and distrust his biases, the anti-rationalist trains himself to trust his faith/emotions/ESP/etc., and ignore scientific evidence. Thus, even when the anti-rationalist feels the intuitive sway of evidence, he or she will strive to ignore it.

BTW, I'm using slash-separated lists in my posts because I'd heard this argument multiple times, from multiple people, each of whom had a different set of ancillary beliefs. Thus, it seems like this worldview is not tied to any particular religion or philosophy.