It seems to me that there might be a fatal flaw for moral relativists simply based on the outcomes, not even just the underlying assumptions or even conclusions.
More specifically, it’s just not fun to be a moral relativist where there isn’t a canonical solution to the problem of what we should principally value because it ultimately produces a gross disorientation with the world (and other humans) which leads to nihilism and to which naturally causes a deep and abiding sense of hopelessness.
And that is not objectively fun, at all. And if that’s kind of the point of all of this, to live life in sheer abundance, then the considerations in and around moral relativism are… well, pointless. Although, thinking about the topic can clearly be instructive.
In any case I think it’s important and necessary to believe in moral primitives that produce outcomes the lead to hope, alignment, and purpose. That just seems a bit more fun, if you ask me.