It doesn't rule out non-local interactions as in general we can rule out local non-hidden variables but not global ones. What these party-pooper theorems do is

It doesn't rule out non-local interactions as in general we can rule out local non-hidden variables but not global ones. What these party-pooper theorems do is largely rule out being able to differentiate non-local interactions from local interactions... which doesn't mean they don't exist, but if they do, there's large pieces of the puzzle missing before we can show they do.