Reb Moshe’s heter seems to have developed over time, but if you learn the teshuvos, especially the third one with all the lomdus, it turns out in the end that one can forget about government inspection altogether. It was a factor in the first teshuva, but once he spelled out his whole shita it no longer seems necessary. Even if there were no inspection at all, the same reasoning would continue to permit all commercially produced milk in a sealed tamper-proof container, so long as it is purchased from a non-Jewish retailer.
To understand this you really need to learn it inside. I don’t personally rely on this shita, but it is not nearly as simplistic as people (both who use it and who don’t) make it out to be. It’s certainly not driven by looking for a kulla. Rather, Reb Moshe held that this is what Chazal really meant all along.