They absolutely give need-based financial aid on a sliding scale based on a collection of household metrics for each student. Now, unlike the Ivy schools that you mention, it's not built into the base price - you actually have to apply for this and end up getting some combination of grant/loan aid offered to you, but close to half of the undergraduate students get some kind of aid.
What I have been told second-hand about why they've never moved to changing this model is that they don't want students choosing Notre Dame from among a list of their elite acceptance options. They want the students that truly want to attend Notre Dame. As such, they do tend to give just enough so that the average accepted student's family can afford it, but they don't want to be considered as a competing option among Duke, Tulane, Dartmouth, Stanford, etc.
One can argue about the merits or even arrogance of this approach, but it's what I've been told.