The manners on hats died when men stopped wearing them in business and formal situations. I think without the requirement of wearing them anymore, dads just didn't teach their sons the rules and it never moved on to the next generations. By the time ball caps became popular, the etiquette was pretty much lost.
Whew MAGA... they don't know the hat etiquette, nor do they know the flag code, and judging by all the terrible things they say about the non-Republican presidents, including drawing them tired up on their pickup trucks, I don't think they know how to respectful at all.
Anyway, about the cycling: no, it's not better elsewhere. In Japan they everyone rides bicycles. Yet, manners are terrible, especially for kids. High schoolers will weave in and out of traffic on their bikes, cut right in front of people driving fast, the works. If a car hits a bicyclist here, it is automatically 100% the car driver's fault, so his insurance takes a hit and he gets a mark on his driver's license, yet avoiding hitting these crazy kids is hard sometimes.
Yeah... it's not only the US.
Well that makes me feel a little better. I'm guessing some of the hat thing goes all the way back to Leviticus and some of the rules that are in that book, but I could be totally wrong. You are likely right that men don't wear them formally like they used to, so ball caps just get lost in the mix.