[dragonwriter]

> As I understand it, the iodine in dairy typically comes from the products used to clean and sanitise the milking and storage equipment (iodophor).

It's possible this has some contribution, but it's not the main source.

> It's not inherent in the dairy itself

Yes, it is; in cows (and humans) mother's milk is where young who aren't eating anything else get their iodine.

> otherwise we could just eat the cow's grass to obtain iodine

Well, except for the fact that grass doesn't work well in the human digestive system, and you'd have to eat a lot of it, sure.