Do we actually need to know how it works, or do we just need to really deeply understand that common sense is not scientific evidence and everything we actually care about can't be predicted just by "Being smart and thinking about it"?

Actually being able to do stuff with Bayes law by hand is going to be not only hard to teach, but probably impossible to remember for those of us who don't actually do math in real life. People forget stuff after a few months or years.

I highly doubt the average person is interested in checking the math on a science paper, so if you want the general public to understand statistics you... have to show us all a reason to, and also teach us all of the related skills needed to make it useful. Or else.... we will all just forget, even with the best teacher in the world.

Most of us aren't doing random game engines as a hobby project or testing things on bacteria cultures.

Maybe they should teach it in context of how to understand a scientific paper, since that's one of the more relevant things for non-pros. If you just teach statistics alone people will say

"Ok, now I know that it's easy to lie to yourself if you don't use any numbers but I don't have collections of large numbers of data points in my life to actually analyze"

"Thinking Fast and Slow" makes a quite extended argument that our brains have a faulty intuition about probabilities, and I nodded along thinking of all the bad decisions I've watched my teams make over the years (either noticed retrospectively, or presaged by myself or some other old hat).

If you have a way to fix this, you would be set for life, going around playing a sort of corduroy jacketed Robin Hood, keeping the rich from stealing from the poor.