If the parents (like OP) have had a couple of small exits, the students (or their parents) pay the full amount.
Edit: At least according to one source, the average student pays 19,000 per year, including fees and housing. And they receive 22,000 in financial aid. Ie, the average student gets a larger discount than what they actually end up paying. https://www.collegesimply.com/colleges/virginia/university-o...
Similarly, Michigan is 18,000 per year with 22,000 in "aid".
Also, both schools are top 100 schools in the US. So even at elite schools, the average student is paying a fraction of the "price".
They get the opportunity to take out more loans, the poorer their parents are. I got a small pell grant a few years, but most of my college was all loans from the student aid office.
Financial aid also includes grants, both school-issued and outside (public and private), and it includes both merit-based aid and need-based aid.
"27% of students received Federal Student Loans averaging $4,835 and 10% of undergrads took out private student loans averaging $11,058."