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Yes, and no.
First and foremost, you need no "justification" for being a decent person. And there are other reasons to be that way, as arbitrary as "I like it this way".
Game theory is strongly related to evolution. It is safe to assume that everything we can observe in nature is a successful strategy. So this confirms the statement: Cooperation is a successful strategy. But the other side of the picture also exists: Betrayal is as well.
What the excerpt omits about the Prisoner's Dilemma (not sure wether it's mentioned in the video, which I did not watch now): The Nash Equilibrium can be the overall worst outcome. What does that mean?
A Nash Equilibrium is a situation in which no player can improve their own position. It is therefore a stable state. Things will change until they have settled in a stable state. It can be shown for Prisoner's Dilemma that the Nash Equilibrium can be the worst case, where each betrays the other. Yes, they would both score better if they cooperated, but the system will still tend towards the state where both play nasty.
When multiple iterations are played, this changes a bit. It seems, if you not just meet once in a lifetime, but can remember your past, and have a common future, it makes more sense to cooperate. But there is still a place for uncooperative exploitation.
So yes, it's true what you say about "best performing strategies", but it should be noted that "evil" strategies don't go extinct either.
It should be questioned how much these theories can be applied to our lifes. I mean questioned, not implying an answer. Either way I find it interesting how behaviour which we associate with morals emerges in very simple and abstract games.