Star-K does not give a hecksher to anything milkig unless it is Halav Yisrael.
The problem with “list” kashrus is policies can change relative to the list. That’s why many people strongly prefer to see a hecksher symbol on the product. Under American law, the company’s can (and do) force the recall of an authorized hecksher, but have no control over an unauthorized or dates use of a company’s trademark.