If I betray people now, then I get a reward now. If I do not betray people, I might get more reward for a long time.
I was thinking more along the lines of a betrayal that would damage the system as a whole, whereas you could use the system for gain if you build it up instead.
There are 4 possible outcomes:
You cooperate, everyone cooperates
You cooperate, someone else betrays
You betray, everyone else cooperates
You betray, someone else betrays.
If there is a built in motivation for betrayal, then human nature assures that you can count on someone else eventually taking advantage of the betrayal action.
This eliminates the 2 options where "everyone else cooperates", leaving only you cooperating and suffering the effects of someone else's betrayal, or you betraying first, and gaining the benefits of the built in motivation for betrayal.
If you are community minded and part of the "systematic human bias towards cooperative behavior" you may choose to cooperate, but it is almost certain that there will be some who are objectively and selfishly purely rational and as such make the betrayal decision to gain what they can before the system is damaged.