> Soggy is not a problem.

It is when you're trying to suck a thick milkshake through one, though...

But usually paper and cardboard that has been in contact with food is not recyclable because it contaminates the batch. That's why pizza boxes also cannot go into the cardboard/paper fraction.

No, that's because pizza boxes are contaminated with fat. That messes up the paper recycling process. Water is fine.

Man, if that's the problem then I can only assume any fast food box is not recyclable too?

The point of paper fast food boxes is not to recycle them but to have no trash in the end as they just burn or rot, all in a sustainable way. In contrast to plastic.