Harvard, or other universities, are not too different from other corporations. Why would they offer to pay for something that their customers themselves would pay for? After all, a Harvard degree is quite valuable, and who's reaping the benefit of it?
They want to make as much money off their endowments (savings) as they can, and spend as little of it as possible on things that they don't need to pay for. Note that while the endowment is huge, actually a large part of it is restricted in what it can be used for, by "generous" donors who put conditions on their money, funnily enough. Turns out that they don't want the money spent on just sending kids to school, they rather want to see their name on art in museums, on professorships, and on department buildings.
Harvard to take an example, has an endowment of $53B, roughly (https://finance.harvard.edu/files/fad/files/fy21_harvard_fin...). Approx. 80% of that has donor restrictions on it and only $9B is useable freely for anything desired.
Last year, Harvard's operating costs were $5B (roughly). To pay for that (roughly):
-- $1B came from student tuition
-- $1B came from funded research grants
-- $2B was released from the endowment
-- $1B in gifts, royalties, licenses, other income
In a sense, $2B spent from such an endowment is not too different from what you would plan for your own retirement. Admittedly, Harvard Management Company (investment arm) earns much better than average though.
It's a choice by these universities, how big they want to be, the ambition of their educational goals, and to some extent, yes their power and influence. They are no different than many corporations.
If you want altruistic behavior and free tuition, well, you want a slightly different system.