I have mixed feelings on this concept.
Public ivy is not a real term -- came from a book /can't remember the title. It's something people have made up to mean "very competitive, high-quality public school" - probably your state's flagship school. Ivy League is a specific group of schools.
You can't know for sure whether your student will qualify for scholarships -- you just have to apply and see how it goes. The criteria for scholarships varies widely: almost all require strong grades, but some require financial need, some require that the student comes from a specific area or plan to study a specific discipline. Some are based on community service or participation in specific clubs. And let's not forget athletics.
Yes, in my experience teaching high school seniors, this is true. I'm thinking of a student of mine from 3-4 years ago: she had a GPA of less than 2.0, so I was surprised when she told me she'd been offered a scholarship from a lackluster private school.
Also, it's wise to be cautious though: some private schools give out freshman-year only scholarships to "hook" new students, and then those students face the choice of paying full price or changing schools.
What I'm reading here is "well rounded high school experience". That's the kid who's most likely to get a scholarship. Grades alone won't do it -- I've known more than one student who's literally never made a B /was shocked that no scholarships materialized. Extra-curriculars, a progression of leadership experiences and community service all make a student's high school experience well-rounded.
Another thing that helps with scholarships -- and it's so obvious: type the application /make sure it's neat and readable, fill the application out completely, and don't bother to apply if you aren't qualified. Seriously, over the years I've been on multiple scholarship committees, and the meetings always start the same way: we go through the applications and reject the incomplete forms. I've literally never been in a scholarship meeting that didn't begin with discarding multiple forms right off the bat.