Moral relativism is invoked almost exclusively by people trying to justify their own actions, after they find that many object to their actions.
Faith has proven itself wholly unsuited as arbitrator of morality. Maybe it's time to give science and logic a go at it.
Any evidence-based system would be better than any system that ignores evidence. We have scientific evidence that the psychology of hate is tangible and specific in how it manifests. We have literally thousands of years of observations and ruminations on this topic, including the teachings of the major world religions.
Culture and society can be surprisingly good at dealing with largely invisible and quite subtle psychology. (Example: love/lust/dating -- Average people are actually quite sophisticated in how they perceive and deal with this.) But societies and cultures are constructed in such a way as to make group psychology around group membership invisible. Group psychology is part of what societies are made out of so the products of group psychology are promulgated as truth.
It's no accident that two different activist movements that are praised for their moral understanding -- those of Gandhi and Martin Luther King Jr. -- were effectively structured around the prevention of pathological group-think and othering. I think a subtle societal understanding of such group-think is the last great barrier to humanity's "Waking up from History."
I think the biggest cause of immorality is that we assume morality cannot be treated logically and scientifically, and instead call it the province of faith. Faith has proven itself wholly unsuited as arbitrator of morality. Maybe it's time to give science and logic a go at it.