I think some are missing the point of this question.
Let's take a product like apple juice. The Talmud never talks about the laws of kosher apple juice, so today kosher certifiers will go into an apple juice factory, figure out what are the odds that the equipment or ingredients have anything non-kosher in them, and certify accordingly. Easy enough. If you bought uncertified apple juice but were somehow able to send it to a lab and absolutely verify it's 100% apple juice, it's kosher! (A hundred years ago in Lithuania, the Aruch HaShulchan actually had this question as there were concerns their olive oil was adulterated. The local pharmacists were able to determine that while it was not 100% olive oil, it was 100% plant material. Which was good enough, kosher-wise.)
The problem here is that the Talmud indicates there was a formal rabbinic ban issued on "milk bought from a non-Jew, unless a Jew was more-or-less present during the milking." 500 years ago if you bought mystery milk from a non-Jewish farmer it was automatically non-kosher. Even if you then were able to manually churn it into butter (pig milk doesn't manually churn), i.e. chemically verify its identity, it doesn't work because of the rabbinic ban. If there was a formal rabbinic ban, we can look for loopholes built within it, but we don't have the rabbinic firepower to actually overturn the ban.
Those who are strict today will insist all milk is covered by this rabbinic ban. (Just as kosher cheese must be made by Jews, even though we're certain this cheese has vegetarian rennet.)
Rabbi Feinstein and others, however, work around this by clarifying the nature of the ban. The ban was not "a Jew must have his eyeballs on the milk at all time", as the kosher supervisor is allowed to step in and out of the barn every so often. Rather, the law was "a Jew must ascertain at the time of milking that it's all kosher." And we ascertain it today via economics and regulation. (Or to take it a step further -- the ban was "don't buy mystery milk from a non-Jew", or better yet, "don't trust a non-Jew with mystery milk." Today's commercially-produced milk in most first-world countries is not "mystery milk", and therefore was not covered by the ban.)
Really halacha only recognizes two categories of milk: "chalav akum", "non-Jewish milk", which was prohibited by rabbinic ban; and "chalav yisrael", "milk that a Jew oversaw." Rabbi Feinstein argues that "chalav yisrael" is defined as "milk that a Jew ascertained", therefore commercial milk is "chalav yisrael." Those to the right argue no, it's "chalav akum." Today we speak of "chalav ha-companies", or "chalav stam", which means: "milk whose status is viewed as chalav yisrael by some, and chalav akum by others."
Those who are lenient could still find room for stringency (as Rabbi Feinstein recommended for elementary-level yeshivas in New York City) as the Talmud mentions that if a piece of meat required a judgment call from a learned scholar about its kashrut, some pious people wouldn't eat it. It's not about "oh what if it's not really kosher?" It's "I don't want to delve into nitty-gritty loopholes to justify my actions."