They count students paying with federal grants and loans, so yes.

A quick reminder that the First Amendment of the Unites States Constitution bars the Federal government from forcing private institutions to participate in chosen political speech. The question of whether you get funding from the Federal Government should not matter, since it's the Federal Government that's constitutionally constrained from doing this.

Of course if we're no longer in a place where the rule of law applies, then sentences like "if it's private and does not get funding..." hardly matter.

Yes. It receives federal grant funding for its faculty. Students receive federal loans for tuition. And international students can receive visas to attend it.