>But I also think, perhaps wrongly, that we can actually learn things about our world, and the way to do that is roughly what we call science. And currently, a lot of people simply cannot think in this way.
I disagree. I think that people contort themselves to find ways to see themselves as internally consistent precisely because they are rational within the inescapable prison of their own perception.
The issue really comes down to causation/credibility. Human psychology will cause each person to strongly favor whatever causal explanations best satisfy their biases.
Your example with religion is a good one. For purposes of this argument, let's assume that event actually did occur and it was not a hallucination.
To a religious person, the event would be a well-deserved manifestation given as a reward for their years of faithful observance, and they would likely continue stronger than ever in their convictions, fully satisfied that their position had been irrefutably proven.
But to a strongly non-religious person, the event you've described would be interpreted as a mental breakdown and cause them to seek medical treatment. After being duly medicated, this person would also likely continue stronger than ever in their convictions, having personally "overcome" a hallucinatory episode.
The same objective event occurred, but the context of one's psychology makes all the difference in its perception. This works with practically everything. No data, no event, no observation exists in a vacuum.
In the case of science/academics, people with agendas and biases are involved every step of the way, from designing the inputs to summarizing and drawing conclusions from the results. And there have been, and surely will be in the future, many significant incidents where the accepted consensus is proven disastrously wrong. Before that happens, the small handful of contrarians attempting to warn about or discuss them are typically considered fringe alarmists, bitter saboteurs, or establishment puppets.
This creates logical space for anyone to disregard the naysayers, set up camp in the contrary position, and gear up for an "I told you so" moment down the road.
If you know this person to be wrong, you will not win by attacking their conclusions, no matter how right you are. Rather, you will win by properly estimating their perspective, understanding why they choose to discredit the particular authority that proves out your position and credit a separate one that doesn't, and then enlightening the situation for them such that their own natural reasoning process will obtain to the "correct" conclusion.
This is called "sales" when it's done on a 1:1 basis and "marketing" when it's done en masse. It can be very difficult when there are powerful entities competing to cast the psychological context in their favor.