I think the reason to cooperate is not to get the best personal outcome, but because you care about the other person.

If you have 100% identical consequentialist values to all other humans, then that means 'cooperation' and 'defection' are both impossible for humans (because they can't be put in PDs). Yet it will still be correct to defect (given that your decision and the other player's decision don't strongly depend on each other) if you ever run into an agent that doesn't share all your values. See The True Prisoner's Dilemma.

This shows that the iterated dilemma and the dilemma-with-common-knowledge-of-rationality allow cooperation (i.e., giving up on your goal to enable someone else to achieve a goal you genuinely don't want them to achieve), whereas loving compassion and shared values merely change goal-content. To properly visualize the PD, you need an actual value conflict -- e.g., imagine you're playing against a serial killer in a hostage negotiation. 'Cooperating' is just an English-language label; the important thing is the game-theoretic structure, which allows that sometimes 'cooperating' looks like letting people die in order to appease a killer's antisocial goals.