NIH falls under HHS, and the HHS acting general counsel was a signatory on the original letter.
That said, affiliated hospitals are not owned or operated by Harvard.
The affiliates could be pushed to drop their affiliation if NIH wanted to play hardball with Harvard.
According to NYT, “ of the $9 billion in federal funding that Harvard receives, with $7 billion going to the university’s 11 affiliated hospitals in Boston and Cambridge, Mass., including Massachusetts General, Boston Children’s Hospital and the Dana-Farber Cancer Institute.”