As the partner of someone who works at Harvard, I have a slight insight into what's happening there. Disclaimer: I may have misunderstood my partner when she described Harvard admissions.
As a rule, foreign students usually pay the full cost of attending Harvard. Then there are legacy students. Many domestic students, primarily low-income or socioeconomically disadvantaged ones, get significant aid, including a full ride.
This is common for foreign students in many countries. They might benefit from a scholarship grant indeed, but that would come from their own government.
The school might have government grants so it can provide better services to all their students, but foreigners usually pay more for the same things.
> Foreign students usually pay the full cost of attending Harvard
That's what I have always understood as well.