[jerf]

Because it's not about the theorems being wrong, it's about people using them in impractical ways.

The Central Limit Theorem is true. Full stop. It can't be wrong. However, in the real world, a lot fewer things are truly Gaussian than may initially meet the eye. It doesn't make the CLT "false", it just means that people who apply it too carelessly are making a mistake. Standard deviation is a thing, but that doesn't make it the right thing for a given task.

A lot of people apply statistics inappropriately. It's hardly their fault, it's basically what they are taught. I remember seeing my wife take her biology statistics courses, which at times seemed to be a course in which you would repeatedly calculate p-values. Just that, over and over; calculate this p-value. Calculate that p-value. Calculate this other p-value. Say "Yes" if it's less than 0.05 and "No" if it's greater. Now do it again. And again. And again. Certainly numbers went in one end of the calculator and came out the other, but did they mean anything? If not, it's not because the p-value isn't "true", just not even remotely as useful as the course was implicitly teaching.

(Yes, words were said about how it wasn't the only useful thing, but the actions spoke loud and clear. Compute p-value. Say yes if below 0.05. Say no if above. Repeat. The current problems all the fields are having with statistics aren't that surprising if you look back to the beginning.)