I updated my post to make clearer, that my response describes NJ.
I also added a note, that there are rare cases where a small town without a high school may have “send agreements” with two different larger towns - in which case the small-town students are privileged with choosing between those two.
One example is a well-off area in Bergen County, another is the case of a town being in a 3 year transition from one agreement to another.
(Similar solutions exist for certain special education which requires a town to bus the occasional special-need student to a most suitable, specialized facility out of town.)
Except for rare cases like this, things are reversed in NJ: you first look at the reputation of towns’ school system, for younger children even the reputation of neighborhood grammar schools, and THEN decide to WHICH town/neighborhood you can afford to move to and afford their school taxes - which sometimes near 5 figures for a nice family home with a good size yard.
Once kids are in college people might move away to towns with less-expensive school systems to save on property tax.