I understood you were defining ‘public’ schools, but you said in the US you don’t get to choose a public school and I was pointing out that sometimes you do. My kids were in 3 different US public school systems (Orange Cty, California; Denver, CO; St. Johns County, FL) and in all 3 they had a public school they could definitely attend because of our home address, but in all 3 districts they had the option to apply for and attend another public schools for no tuition. Some of the schools were very hard to get accepted to while others would love to have more students (schools get more money from the district/state if they have more students). Usually, the student has to provide his own transportation to that school (in Denver they get a city bus pass) so that can really limit where the kids can go.
I don’t know what kind of choice those in NJ have to enroll in a public school in the same district or in another district.