So you are basically saying that sometimes it is in your own self-interest ("own utility") to cooperate with other people. Sure, that's a pretty obvious observation. I still don't see how it leads to utilitarianism.
If you terminal value is still self-interest but it so happens that there is a side-effect of increasing other people's utility -- that doesn't look like utilitarianism to me.
I was only trying to make the obvious observation.
Just trying to satisfy your empathy does not really look like pure utilitarianism either.
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.