80% of American households earn less than $100k and if 80% of the students got their education for free that wouldn't work out well for the universities.
Actually, it would work out just fine (for top-tier universities). They can easily survive on the investment returns from their endowments, along with donations from wealthy alumni.
Also, removing the tuition would make them look cheap to their target audience. In reality, a significant majority of students do end up getting financial aid. It just has to be indirect, so it doesn't harm the school's reputation.
I don't know anything about CMU's financial aid system, but in the Ivies, this situation certainly would not have occurred.