That the school system sucks is not an unavoidable fact of life, it's a problem that needs to be fixed. I admit, another problem on the big pile of problems that need to be fixed, apparently.
They have both systems. And public system always sucks, because there isn't an unlimited amount of good teachers for all the public offer and, not the same amount of resources. Private schools have better everything because there's competition.
Not true. There are plenty of public systems that work very well, and better than private systems. Just look at health care systems. Schools could be the same, but you have to invest in them. Plenty of countries have excellent public schools.
State level funding is a big one, poor areas get the same funding for local schools as wealthy areas. I don't understand why "progressives" in the US don't support a more egalitarian funding model, it seems like a very achievable goal and compensates for quite a few privileges.