Let's turn that around 180 degrees - without statistical controls how can you decide that a box did it?

You have no idea, you just like the box so you declare "the box is the king". It sounds good - the box has something to do with babies after all, so let's assign all the improvement to the box.

Sounds ridiculous doesn't it?

If you want to say "I have no idea whatsoever what caused the improvement", then I'll grant you your statistical rebuttal. But if you want to say "The box did it", then your statistical rebuttal is disingenuous.