The story of R’ Moshe throwing up is pure fabrication. I have a friend who is close with R’ Dovid Feinstein, and I asked him to find out the truth behind it. He asked R’ Dovid, who laughed. He said, “If my father threw up the milk, it must have been spoiled! He wouldn’t vomit from it otherwise.”
R’ Dovid is not makpid on chalav Yisrael, despite what Joseph may claim.
I also have a neighbor who lived right across the street from R’ Moshe when he was young. He’s also one of the few people in my community who are makpid on chalav Yisrael. When his father was visiting for a Shabbos, I once asked him why. He said that R’ Moshe told him personally that he considered him to be a baal nefesh, and that he should be machmir. He didn’t say anything about shas hadchak.