Most private schools require CSS Profile from students applying for financial aid. It asks for much more financial detail than FAFSA. Each school has their own proprietary formula for calculated need based aid on the info from CSS Profile.
Most public schools use FAFSA only to determine need, but most schools that use FAFSA only do not meet full need.
So do not think of your FAFSA EFC as what you will pay at one of those schools, in general you will be expected to pay more than your EFC, sometimes significantly so. When schools do this, it is called ‘gapping’.
Make sure to run the net price calculators at all schools your kid is considering. Many are highly accurate, unless you are divorced, own a business, or have real estate beyond a primary home.