I do think Bayes is an improvement over the frequentist methods. But I do think that it's not the main place where people go wrong.

The main failure I see is the "What a coincidence!" failure. People notice an event that is a one in a billion chance. They say to themselves "Wow ! There was only one chance in a billion for that to happen by chance. This means there must be something special about me/the universe/a higher power/whatever."

They completely neglect two other numbers.

1. The number of events that occur every day. You get 86400 seconds every day, and the proportion of those when you're awake is a time when something unlikely could occur.
2. The number of other possible unlikely events that you would also have noticed. This is a pretty big number in its own right.

Once you've taken these into account, you discover that you actually ought to expect to experience a steady flow of apparently coincidental unlikely events throughout your life. It would be notable if it didn't happen.

I suspect this is the most important step for most people - the principle of thinking numerically about the things that happen in an orderly way. Going on from there to Bayes' theorem is then a matter of being taught how to do the maths correctly.

It might be a good idea to put Bayes' Theorem into English (I haven't seen that anywhere). My attempt would be this:

To judge the hypothesis H given the evidence e, you need to know how well H explains e AND the base-rate of H AND compare that with all alternative hypotheses.

So even if you don't have any numbers to plug into Bayes' Theorem, the theorem is still a nerdy reminder not to neglect base-rate and to think of alternative explanations.

Thinking is itself an act that modifies your subjective probabilities - whatever reference class you pick."...persuasion is powerfully affected by the amount of self-talk that occurs in response to a message.[2] The degree to which the self-talk supports the message and the confidence that recipients express in the validity of that self-talk further support the cognitive response model."