World War II seems to have some possible examples. Many Jews who didn't think about escaping until it was too late, because it was too horrible to believe. Neville Chamberlain thinking he could appease the war away. Possibly Stalin's reaction to Hitler's invasion:
Accounts by Nikita Khrushchev and Anastas Mikoyan claim that, after the invasion, Stalin retreated to his dacha in despair for several days and did not participate in leadership decisions.[145] However, some documentary evidence of orders given by Stalin contradicts these accounts, leading some historians to speculate that Khrushchev's account is inaccurate.
Hitler's delusions of a German victory toward the end of the war. The Japanese holdouts.
These are the first things I thought of, anyway, although it now occurs to me that comparing your audience to Hitler would have some rhetorical drawbacks.
I think we have to be careful to avoid hindsight bias when thinking of examples of this. For example, it is quite possible that Jews who chose not to leave Nazi Germany before Kristalnacht were in fact acting perfectly rationally. Certainly Weinberg makes a reasonable case that they were acting rationally in this essay (and yes, I realise that he has plenty of reasons to try to justify himself, but that doesn't change the fact that it's possible he made a sensible decision).
I'm looking for historical examples of "flinching away," so I can illustrate the concept to others and talk about motivated cognition and leaving a line of retreat and so on.
The ideal example would be one of motivated skepticism with grave consequences. Like, a military commander who shied away from believing certain reports because they implied something huge and scary was about to happen, and then the huge and scary thing happened and caused great damage. Something like that.
What examples can you think of?