It would seem to me that a self-interested person would recognize that cooperation among and with many others satisfies that self-interest. 
Yes. It is, well, selfish.
If you think self interest is the same as Nature Herself, then you end up, as Hobbes put it in Leviathan, with human life as “solitary, poor, nasty, brutish, and short.”
Everyone for himself, and the devil take the hindmost. If you want something, take it, and if the other fellow objects, kill him. Contrary to expectations, this kind of attitude does not lead to great happiness.
It would seem to me that a self-interested person would recognize that cooperation among and with many others satisfies that self-interest.
“If you think self interest is the same as Nature Herself”
Who does that? What does that mean?
“Everyone for himself, and the devil take the hindmost. If you want something, take it, and if the other fellow objects, kill him. Contrary to expectations, this kind of attitude does not lead to great happiness.”
No, but egoism does.
Of course, what you desribe has nothing to do with egoism as the author describes. Even though the author points out the common error of confusing egoism with egotism and hedonism, you still make that mistake.