Public education is very focused on drilling particular, objectively-measurable skills. These are often very valuable -- basic literacy and numeracy are obviously hugely important. If you pay attention through high school you come out with a better grasp of politics, history, various sciences, and literary analysis than 90% of the population. You can be quite successful without any of this stuff, of course, so it depends what you mean by "important". A lot of people feel that schools don't do much to teach "critical thinking", "how to think", etc. And that's probably right. But it's worth considering that this is very hard stuff to teach, and often has prerequisites.
Like, it's very hard to teach someone to think about history without first teaching them a lot of history. You could instead just teach your conclusions, but that's not actually going to teach people to think for themselves and is often going to just be propaganda. There's a ton of annoying prep work that you have to do before you're remotely ready to start drawing interesting conclusions from the facts, because you need ready access to all the facts. If you want to have anything interesting to say about biology you've got to cut open a lot of frogs.
But it's also just hard to teach critical thinking full stop. Science classes, even with labs, are really bad at teaching kids to do science. A lab session is still generally a guided tour of an experiment that leaves very little room for theorizing and hypothesizing and really experimenting. But think about what you'd have to do to actually give students this. If a teacher is supposed to handle 20 students at a time you really just can't do it. There's a reason why we find that schools matter a lot less than parental involvement at the top end.
