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One thing the pandemic taught us is that school is primarily baby-sitting for parents that work.

Maybe not "extremely" important, but autodidacticism can be terribly socially pathological, which pedagogy is meant to deal with. A lot of the point seems to be social, we want our kids to be able to navigate each others ideas with a common logic

Then why do we put them in school where in most cases they barely get to interact in that way?

What do you mean?

> For example, students who don't learn any math until 5th grade caught up to their peers by 6th grade:

That was a study done from almost a hundred years ago, its hard to imagine its exactly applicable today.