You get at the heart of what makes the prisoners dilemma interesting.
When you play the game once, betrayal makes sense. Yes, cooperation might be better but the juice isn't worth the squeeze. No consequences, tragedy of the commons, etc
If you play on repeat, you need a strategy. It generally comes down to: Do you penalize someone for betraying you?
Like you said, if the game is repeatedly for infinity, then cooperation makes the most sense. Betrayals are punished at a cost higher than the betrayers reward. Altruism prevails.
if the game is played on repeat until a known endpoint, then it gets complicated. It makes sense to betray the last game: you can't be punished for betraying on that last game. And even if you don't want to betray, strategically the other player also knows the betrayer has an edge in this case if they don't also betray. So the last game should be a double betrayal.
And with that leap, it also makes sense to betray in the second to last game, since there's only a gain if you betray without being betrayed, and the last game's outcome is already double betrayal. Now the logic is betray the last two games. And so on, etc, and the whole thing unzips.
Now cooperation is irrational and selfishness prevails.
Thus, altruism works when the elders plant trees to make shade they won't feel. But if one side thinks the game will end, or fears the other does, then the whole thing falls apart.
I said this when the sale first happened. Those billionaire Saudis didn't loan him the 8 bil needed to finance the sale, they gave it to him as payment for a job. And it's 100% the long tail of the Arab spring reaction