No public schools "feed into a top college." I went to a public HS regularly ranked #1 in the country, and students went to exactly the colleges you'd expect based on their SAT/GPA.
Ivy League school admission offices literally have lists of preferred schools to recruit at, and give more respect to GPAs at those schools, and many of those schools are public schools. I attended one.
I'd imagine students at lower ranked schools had lower SATs scores, and thus didn't get to similar colleges.
Lot cheaper to spend a grand in an SAT prep course than buy a house in a "good school district" hoping it has an indirect benefit.
I'd imagine the difference between a highly and a lowly ranked school is more than an SAT prep course.