Most universities have two general types of graduate students (not counting MD and law). Some students (business school, for example, but also of course MD and law) actually pay the grad tuition at the same rate being waived for those who are on an assitantship. The tuition-paying students make the calculation that they will be sufficiently better of with the degree as to justify the expense.
If the Yale grad student union wants its members to be classified as employees, then they should not receive educational credit (a degree) for the work they do for that pay.
When NSF (for sure, not so sure about NIH) budgets a grant award, it is for the total direct + indirect, not for direct only, with the university not getting ‘extra’ for the indirect. Tuition being charged to the grant is a 1-for-1 reduction in the money available to do the research. I’m not sure how often in practice tuition is charged to grants in the PhD programs in med schools, but in Arts & Science-type departments much of the time the school is willing to cover the tuition waiver themselves.