Neither does the US: foreign students are ineligible for federal grant money and are much more likely to be paying full tuition than US students. The majority of federal moneys given to Harvard go to their hospital programs, with the remainder to "research" (a somewhat frustratingly ambiguous term). There has been zero evidence presented that foreign students there pose any particular security threat against the US. Given the effect such a smoking gun would have on perceptions of this matter (and beyond), one can't help but think no such evidence exists.

It's a bit obvious Trump just picked a target to attack that will resonate with certain anti-intellectual segments of his base of support (and given his rather infantile, petty outlook, punish an institution that rejected his kid's application). I suspect he also wants to quell criticism of his buddy Netanyahu's policies and actions in Gaza (and, to be fair, address some campus protests at other institutions that have hedged into legitimately antisemitic territory...although I don't know that any at Harvard that went too far).