That's all well and good, the problem solving part.
The not being able to write or speak is a problem, the inability to do math and science, another problem, lack of knowledge of history, and the political process, another problem.
About the hats - trying to be nice here..really am. It's something about respect. If the school sets it as a rule and you let students wear them because you think it's a stupid rule -- then you are teaching them insubordination at an early age. Respect by allowing for insubordination is not the kind of respect you want and you aren't doing them any favors. You are damaging them in the long run. I hope you see it someday.
Following a set of standards and insisting on excellence does not hinder problem solving. It helps form a habit and a discipline so that a student will be ready to go out and solve the problems of the world, or give their talents to the world and will be of use to the world.
I am not an educational philosopher and I will not give advice to parents - but if someone wanted to know what I might think -
Unless we have already arrived at where we want to be, then we should be preparing for the next step.
A primary schooler should be preparing for intermediate levels. An intermediate student should be looking at Junior High who should be preparing for high school who should be preparing for adulthood - college - trade school - or work, and it goes on and on.
Accepting rules that one doesn't like or think are stupid is a fact of life in the real world. Hats are stupid in class? Maybe eye protection being required for every time someone operates a tool on a construction site will seem like a stupid rule.
Logic. Reason. It really does go a long way. : )
Insubordination -- well in my experience, it doesn't go a long way. I've walked on the edge of it myself a few times and was lucky I didn't lose my job.