Potatoes are a New World crop. Medieval European peasants didn't have them.
No, you wouldn't, as a European of any social class, except maybe in the very late middle ages at the boundary of the Renaissance; potatoes were a New World import in the 16th century, which might just still be considered the Middle Ages in some parts of Europe.
It was an important source of protein in centuries past, and was traded all over Europe. It would've also been an effective way for inland residents to get iodine.
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.