At my employer in the US, sick leave was drawn from your vacation days... hence very sick people always coming into the office to work.

That’s common. School teachers face the same problem. They don’t quality for sick leave until they’ve used up all their PTO first.

My father, a high school teacher, has some absurd >1000 day sick day balance right now. He keeps putting off retirement for some unknown reason despite complaining about how terrible the system is, and a common response from my siblings is "maybe you should just be sick for three years…"

I suspect this varies by state, but the idea that sick leave for teachers is lacking has never been a common theme for any of the teachers in my family.

There's been a general shift in the US from a system where large organizations had separate pools for sick time and vacation to a common PTO pool. But you still see organizations, either because of inertia or union contracts, still have the prior system in place. And, yeah, the traditional system tended to have more sick days than anyone would use outside of serious illness or injury although, as I recall, there usually wasn't carry-over.

The US is an exception though, since having any kind of leave at all is a gift of of the employer and not defined in law.

For the rest of the world, the idea that an employer can arbitrarily decide it is not possible for someone to be ill more than a certain number of days per year is bonkers.