>Okay, so shame exists because it is good for a society that is built on shame?

Yes. Shame exists because it gave some evolutionary benefits to the primates who had it, like being able to live in a society and cooperate and don't cheat/kill/fuck over each other as much (if you think we do bad stuff too much, wait till you see what we can do without shame).

>I believe there are better ways to build relationships with others

The problem is that made-up (e.g. of our own making, requiring us to think about them and follow them rationally), not instinctual, ways, are none effective at all compared to innate, evolutionary, feelings like shame...