Harvard in particular is so wealthy it always charges tuition based on ability to pay. The poor who go to Harvard pay nothing. But, of course, the poor seldom go to Harvard because they aren't admitted.
Just saying. There are mid-range school where, all else being equal, your scenario might be true. But those school depend on those same lower income students and, well, I'd be in favor of them buckling and charging less.
If underprivileged kids represent anything above about 5% at Ivy's I would be shocked. One stat from Boston Globe was that only 3% of Harvard comes from the US bottom 20% economically.
The Ivys do more to perpetuate the existing power dynamic than they do to raise anyone up. They are far too small and far too nepotistic to do otherwise.
If you want to raise the middle/bottom up, you need to look at the large state university systems which are orders of magnitude larger.
Also there are lots and lots of programs state and private to fund low income higher Ed. It is not a dichotomy.