Rav Moshe’s heter is that it is possible that American milk is not cholov stam but cholov yisroel. If you hold that it is permitted, you don’t need a heter. CY is not a chumrah, it is a l’chatchilah, and there is a big difference.
The Gezeirah was that a Jew has to supervise the milk, not that you have to have some kind of insurance that it is Kosher. Rav Moshe’s heter – however far one takes it – is to interpret the Gezeirah that any kind of insurance is sufficient to fulfill Chazal’s gezeirah, which would make American milk cholov yisroel. But that itself is a matter of interpretation. If the Gezeirah was simply that you need a Jew supervising the milk, then even Rav Moshe would concede that even if you have reason to be comfortable that the milk is kosher, you still did not fulfill the gezeirah. In addition, even if you will accept Rav Moshe’s interpretation that the gezeirah is only that you need insurance not specifically Jewish supervision, who says that the laws are sufficient insurance that would satisfy Chazal? If the legal penalty for violaitng the government’s milk laws is a fine, but the company will make more profit by violating the law, then how do we know that law is insurance? Very very often stores violate the Kosher Consumer laws. They put treif meat in the Kosher section etc, and they get fines that do not deter them from being repeat offenders. Then there is the question as to who says the laws are being honestly enforced? As a certain godol said about the heter of the inspectors: “You give me two hundred dollars and I’ll give you four inspectors.” What halachic basis is there to believe the inspectors are doing their jobs, being that Akum have no ne’emanus halachicly?