Joseph-The OU is not involved in making sure the milk is from a cow, what they do is have yotzi vnichnas by the dairy to make sure the milk does not get unkosher ingredients (such as non-kosher vitamins) added in.
But you actually bring up an excellent point, that I have not thought of before. If the OU is inspecting the dairy, then acc. to Rav Moshe tz”l, it could possibly suffice. Again, Rav Moshe holds the gezeriah only takes effect when it reaches the hand of a Jew (IM YD 1:49). Therefore while it is in the dairy, why can’t the OU’s yotzi vnichnas suffice to actually make it chalav yisrael without government inspection needed? Unless you say that the OU would have to be tomid at the dairy for this to work.