I think one reason for the lack of concern with the reason for the sick day is because it is tied in with holiday/vacation time. Every year, I get 2 weeks for vacation, 1 week of PTO/sick days, and we get off the minimal holidays. For example, New Year's Day is a holiday, but Columbus Day isn't. But, in truth, vacation, sick, PTO, holiday time is just put into one field. So, any day I don't work, 8 hours is taken out of that bank, regardless of the reason (and I get paid).
Every employer I've worked for either has you lose all unused leave at the end of the calendar year. Or, there is a bank that just keeps accumulating. With my last employer I got a check when I quit equal to one month's salary because I'd had that accumulated. With my current, I have about 150 hours saved. I've traveled extensively before, but right now I just want to be a homebody.
Before my father retired, he had 4 weeks of vacation plus holidays plus unlimited sick days. He really didn't know what to do with more than two weeks. In fact, he started to work 4 day weeks in the fall just to use the extra two weeks.
My personal belief for the difference is that in America the focus is more on the home. Money is more likely to be spent on an upgrade for the home, than a nice vacation. So, more than what is needed to rest every now and then isn't missed.