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.