It is not quite correct for everyone to be quoting Rav Moshe, as though the permissibility of drinking USFDA milk originated with his responsa on the subject. Rather, most American Jews were drinking what he called chalav hacompanies well before RMF arrived in the US. They had rabbanim who had already ruled it was permissible, such as R' Dov Revel, R' Yisrael Avraham Abba Krieger, (among other greats of early 20th century American Judaism whom time has forgotten because the post-War immigrants never met them) R' Breuer, R' Moshe Soloveitchik and his son R' JB Soloveitchik. In the early days of Lakewoodthey served regular milk and Breakstone cottage cheese. Yes, that stopped when CY became more readily available, but obviously the yeshiva wouldn't have served it had R' Aharon Kotler believed CY was mandatory. (They could have sent someone to a nearby farm -- Lakewood was near farmland back then.)
Then there were those who didn't permit USFDA inspected milk. Those who immigrated in that same post-war wave were therefore confused which to do. RMF sided with those who permit, giving the version of the justification he found valid. But the leniency doesn't entirely rest on his shoulders.
Also, it's clear Rav Moshe's language shifted as chalav Yisrael (CY) became more available, in the earliest responsum treating CY as a stringency above the baseline (Irgros Moshe YD 1:47-49), the middle more equivocal (2:31,35) and the latest (4:5) more like not insisting on CY being a leniency. However, Rebbetzin Feinstein did not observe chalav yisrael, and it is well known in their community that Rav Dovid Feinstein to this day drinks what his father zt"l labeled "chalav ha-companies" (company milk, USFDA approved). So I wouldn't read that much into the change in language, if it didn't impact what he told his own family! In practice, he treated CY as a personal stringency, even to his last day.
The basic conceptual division is in how we understand CY:
According to the Peri Chadash (Yoreh Deah 115:6, based on the Radbaz), the risk of consuming milk adulterated with non-kosher milk is like any other kashrus risk. The rabbis identified it, and told you to avoid the risk by drinking CY. But if you could take care of the kashrus risk some other way, that would be just as good. And so milk subject to government agency rules, like the USFDA, would be kosher.
The Chasam Sofer (responsa, Yoreh Deah 107) rejects the Peri Chadash's position, believing that CY is a distinct rabbinic legislation.
Rav Moshe's position is conceptually based on the Chasam Sofer, but ends up closer to the Peri Chadash in implementation. He writes that there is indeed a rabbinic law requiring a Jewish observer, but "observation" boils down to knowledge. And Rav Moshe brings other examples where the law uses the term "sight" but knowing in other ways is sufficient. Therefore, as long as one knows for sure (or sure enough) that the milk isn't adulterated, the rabbinic law of Chalav Yisrael is being obeyed.
In the Arukh haShulchan's analysis (YD 115:1-17) we have both issues:
1- Eliminating the risk of drinking milk that is adulterated with something that isn't kosher.
2- A piece of rabbinic legislation that requires some kind of Jewish observation during milking. The legislation may have been motivated by #1, it is still legislation that would require a greater Sanhedrin to overturn (115:16). And we don't have what it takes to make any Sanhedrin. (And besides, the AhS adds, for every reason given for an enactment, there are numerous unstated reasons.)
If the farmer doesn't own any non-kosher animals, and adulterating with non-kosher milk would make the milk more expensive instead of less; or the farm is focused on butter production and adulterating would also make no sense -- we took care of #1, but the Arukh haShulchan (and those who obligate CY in general) would still require Jewish observation. (115:9-10)
However, in such a case where the risk of non-kosher is ignorable, this observation could be minimal. He could come in during the milking of the last cow of the run and all would still be kosher. This is not the random spot check inspection (yotzei venichnas) of kashrus, but meeting a separate requirement.
And the Arukh haShulchan considers utensils used for cooking non-CY milk to also be non-kosher. (115:11) But he is lenient on butter (115:20 onward), noting that the gemara only mentions milk and cheese, not milk products in general. He therefore would presumably permit the vast majority of manufactured products as well, as it is far more common to use milk powder than actual milk.
Note that the AhS is known for representing accepted practice in Lithuania. However, the Polish government didn't offer consumer protection of the sort the USFDA (or parallel contemporary government entities) provides. The specific question Rav Moshe Feinstein discussed was purely theoretical. Still, I am not sure there is room for Rav Moshe's leniency in the Arukh haShulchan's formulation, despite the fact that both are understandings of the Chasam Sofer's position, and both have similar background in terms of accepted practice and custom.
I just want to provide a balanced presentation, rather than looking at one side to the exclusion of the other.