American universities are in demand among foreigners because they are a gateway to network and life in the US.

If it's a choice between say Princeton in the US and Harvard outside the US, Princeton will be the choice for many.

The universities could have expanded campuses and opened up new ones at any time. Harvard could easily double or triple their class size with no negative effects in terms of student body. They choose not to because it would dilute their brand and they're more about an exclusive club than educating students or doing research. Their doner class that finds their hedge fund doesn't like that.

So let's not pretend these institutions are noble