The Shulchan Aruch does not mention government supervision.
The poskim were mattir it to people who asked. BMG bought milk from regular companies, when Reb Aaron Kotler was in charge. In fact, the Yekkishe bochurim arranged for Chalav Yisroel from a kibbutz Hachshara in NJ. The first delivery arrived on Shabbos, leading Reb Aaron to comment sharply about those whose chumros caused them to forget basic Halacha. Reb Yankel Schiff is still alive, and he is supposedly the source of this story.
You are distorting the argument. Rabbonim were meikil, not people who worked on Shabbos. It was accepted as a דבר המותר based on Rabbonim who didn't write Teshuvos. Ba'alei Teshuva and Geirim don't know this, because they have no connection to past generations. But Reb Moshe did, and he explained the heter they relied on, in his opinion.
At the time, assering milk from the companies is like those guys who tried to claim that all eggs are ossur, because of the leghorn chicken something something. Reb Moshe refused to countenance these kinds of claims - if doros of ehrliche Yidden do something, it is itself a source lehalacha. (Reb Moshe was very consistent about this Shitta, regarding Techumin on Shabbos, covering ladies' hair with a sheitel that looks like hair, shaving beards with a machine, and what beracha to make on a pretzel)