> Cooperation seems to be just as fundamental, if not more, than competition
Both are fundamental. You can't survive without cooperating, but you also can't survive if you try to cooperate with the entire biosphere because ultimately there is competition for scarce resources. If you don't assert yourself to claim your share of those, something else will.
Yes, but does your kidney compete with your lung?
I would highly recommend you read through the paper. Alongside these "advanced in individuality" comes reduced (internal) competition.
I agree that both are important but competition only seems to be important at the very edges of what selection is acting upon while cooperation is truly "fundamental"