Not sure why this wasn't mentioned more often but it's because of donor money. Donor money that funds things like new cafeterias and other facilities ultimately benefits everyone at school and helps keep tuition prices in check, also complicates the ethics ie if they didn't have enough donors, then tuition will go up and it'll be even less affordable
I think the argument rings hollow to me mostly because it's not that Harvard has to charge this much. I'm sure they could be providing the same quality of education for 1/5th the price. In fact, with the endowments many of these schools have, they could probably go tuition-free for a couple of decades and still be fine.
They charge this much essentially because they can (govt-subsidized loans), and because it helps them maintain a certain reputation.