Why were schools so against wearing hats? what's wrong with a hat?

Teacher here in urban environment.
Reasons for no hats

1. A million ways to cheat using hat.
2. Can be used as a weapon
3. Blocks view of other students
4. Distracts other students
5. Easiest way to pass bugs

All your reasons are way too intellectual to be a reason the government does anything.

It's a leftover from Christian tradition.

1 Corintians 11: 4 - 7 4 Every man who prays or prophesies with his head covered dishonors his head. 5 But every woman who prays or prophesies with her head uncovered dishonors her head—it is the same as having her head shaved. 6 For if a woman does not cover her head, she might as well have her hair cut off; but if it is a disgrace for a woman to have her hair cut off or her head shaved, then she should cover her head. 7 A man ought not to cover his head,[b] since he is the image and glory of God; but woman is the glory of man. 8

Schools are all about conformity and enforcing arbitrary rules. Schools are designed to make you docile and get used to bullshit so you're less likely to revolt in your job.

My theory is that modern (post industrialization) schools fundamental purpose in society is not to educate but to prepare a workforce. They acclimate people into accepting very arbitrary rules that will be normative in the work place.

There's a lot written on this and a fair amount of disagreement. Here is a decent high level / non academic article. https://qz.com/1314814/universal-education-was-first-promoted-by-industrialists-who-wanted-docile-factory-workers

Only reasons I can guess:

Traditionally, it's considered disrespectful to wear hats inside. Why that tradition exists, I'm not sure, but it's one of those etiquette things, like not putting your elbows on the table.

It could also be along the same lines as hoods - obscuring your face and making policing the school harder.

Students don't get to be warm. Only teachers get to be warm.