Assuming you mean viral load referring to mucuses or blood of the infected. Given that the human influenza virus’ entry requires the viral surface proteins (hemagglutinin (HA)) to bind to acids present on respiratory epithelial cells along with cleavage of HA by host cell proteases (enzymes that breakdown proteins) to facilitate membrane fusion. These trypsin-like proteases are mainly expressed in airway tissues, restricting influenza viral tissue response to the respiratory tract. I would say it would be highly unlikely for influenza viral replication existing in an environment lacking this crucial interaction let alone a low-pH environment like the GI tract

That’s a fascinating question. I’d love to hear from an actual epidemiologist.

You don’t swallow all of the milk and thus not all of the virus particles. There’s always some residue that can then start to infect you just as if you inhaled the virus.