It entails that behavior that people consider moral, tends towards having the property that if everyone behaved like that, things would be good
This is just circular. What is “good”?
Rule of law, equality before the law, Rawlsian veil of ignorance, stare decisis, equality of opportunity, the golden rule, liberty, etc. Generally, norms that are symmetric across space, time, context, and person. (Not saying we actually have these things, or that “most people” explicitly think these things are good, just that people tend to update in favor of these things.)
Evidence that “most people” update in favor of these things? It seems like a very current western morality centric view, and you could probably get people to update in the opposite direction (and they did, many times in history).
Why are you personally attacking me for discussing the topic at hand? I’m discussing human nature and giving myself as a counter-example, but I clearly meant that it applies to everyone in different ways. I will avoid personal examples since some people have a hard time understanding. I believe you are ironically proving my point by signaling against me based on my beliefs which you dislike.