This is a letter of his published without the context or the even the question it was in response to. Is that printed anywhere else? The one on the next page is actually one of the letters I was referring to written to Rabbi Ochs in Toronto and was published in other places as well.
I was not able to find the actual question but a bit of detective work should give us plenty of context. The question was asked by a Rav Avrohom Weinfeld of Monsey, mechaber sefer Lev Avraham.( He is the shoel in another teshuva in the Igros.) In Lev Avrohom siman 74 he strongly opines that cholov stam is assur in general, and he specifically calls out the schools that are serving it to their talmidim. He then proceeds to bring several teshovos from Rabbanim that replied to him, concurring that it is assur, among them Rav Yonason Shteif.
The teshuvos of the Rabbonim are all dated parshas shlach תשטז, the exact date that Rav Moshes reply to him is dated, so we can reasonable assume that the letter he sent to Rav Moshe was a carbon copy of the one he sent to the other Rabbanim. Interestingly he left Rav Moshe’s reply out of his teshuva, presumably because it did not corroborate his position, since Rav Moshe did not outright assur Chalav Stam.
Rav Moshe does not seem to be limiting his reply to yeshivas, and indeed Rabbi Weinfeld was opining that it should be assur across the board to which Rav Moshe was replying. I think it’s fair to say there is a strongly worded opinion from Rav Moshe that everyone who has access to C”Y, even at additional expense and inconvenience, ought to be machmir.