On the other hand, I think people can acquire a pretty good ability to recognize fallacies without a formal understanding of what a good proof is.
I just feel there is a difference between a "fallacy enthusiast" (someone who knows lists of logical fallacies, can spot them, etc.) and a "mathematician" (who realizes a 'logical fallacy' is just 'not a tautology'), in terms of being able to "regenerate the understanding."
This is similar to how you can try to explain to lawyers how they should update their beliefs in particular cases as new evidence comes to light, but to really get them to understand, you have to show them a general method:
(Yes, belief propagation was more or less invented in 1913 by a lawyer.)