I think it's more nuanced than that, and it really depends on the level of education.

Making kids memorise things also teaches them the process of learning a thing. Testing them on facts, dates, and formulas has value because it tests whether they're able to learn those facts, dates, and formulas.

In high school maths, I had to learn formulas. When I was applying to university, the admissions test came with a formula booklet. It was assumed I knew how to learn formulas, they were testing whether I'd learned how to look up the correct formula, and apply it. They weren't just testing my mathematical ability, they were simultaneously testing my reference skills. I only really appreciated that when I was much older.

Memorization is such a strange thing to try to teach. I was never good at that in school but could sing any song on the radio after hearing it once. In school I was good enough at math up to a point because I was so bad at memorization got good at thinking my way through it. Was much better at word problems than equations in elementary school.

I do agree with the premise of the shower thought - part of being educated is learning through mistakes. Making mistakes is one of the fastest ways to learn something, and is the main reason I'm good at my job. I am happy to work in accounting where mistakes don't kill anyone.

There was a popular author of technical books about the Commodore 64 who thought nuclear bombs aren't real.

You can have a few neurons really good at one thing, they don't map over to other things.

Education improves intelligence marginally by forcing people to use parts of their brains more, which then remodel to be more efficient.

Most of your intelligence is set.