I interpret it to mean that we shouldn't be adding sugar to any of our foods. The natural sugar in the foods we eat is plenty.

I wouldn't even go that far. A slice of birthday cake won't kill someone. Added sugars have their place, but we shouldn't be adding them where they aren't needed and we should consume them in moderation. It's wild how much random stuff has added sugar. I've even seen deli meat with added sugar. Who is asking for corn syrup to be pumped into their roasted turkey?