That's actually not true. Foreign students generally don't get aid and have to pay full price. As far as diversity is concerned, what matters to colleges is racial diversity. that is, they want to have a large non-white, non-Asian percentage in the student body, so for this purpose, an American black or Latino student is just as good as a foreign one.
Actually, it is true.
I work with 23 colleges. In conversations with administration of all but the public college and universities there is a great push towards getting foreign students. They are given preferential treatment.
In fact, the reason why a lot of “colleges” are changing to universities is to attract more of this foreign diversity. Perhaps the dozen or more institutions that pay my salary are unusual. But because of the vast difference in socio-econmic and concentrations (my words are chosen carefully) I think it might be more accurate than a random position.