It doesn't look to me like it works as a description of morality in a general sense.
It doesn't explain the idea of moral progress. Or why people listen to unpopular moral reformers. Or why people even make judgments about morality. (if people truly believed that morality was relative, then they would believe that others couldn't be wrong- if morals are relative, then the others aren't wrong. They just hold different views that are right... from the others' perspective. If they accept that the others are right, then by why would they care what they think?). — anonymous66
As previously stated, I don't very much care for cultural relativism, though this is more because of the "cultural" part, as there are subcultures and radical individuals that the theory does not account for to be considered complete in my eyes, as well as other problems like viewing morality as purely social. Rather, it only works in the language of a sociologist, describing what groups of people consider good behavior. Since what I will say applies to the majority of nonobjective meta-ethical stances, I will stop referring to cultural relativism specifically and talk about these positions more generally.
First, none of nonobjective moral positions I am aware of state that everyone will accept the truth of their position. These positions can hold the majority of people will believe morality is objective. Second, compare morality to a similar concept in the study of value theory: aesthetics. There are a lot of groups of people who argue that aesthetic qualities are objective (in some sense of the word, "objective"), that a particular painting is objectively bad and that people ought to respond to it as bad. Beauty is a real objective property that certain objects possess. Now, there are people who dispute this notion; they claim that aesthetic values are ultimately subjective. They may say that, due to our shared nature as human beings, that we tend to find certain things pleasant, but ultimately what we get out of aesthetics is subjective. Certain people have certain tastes and that is all there is to it.
This does not stop these people from arguing over aesthetics, saying "that was a bad movie", arguing against paintings in their dorm room because they find these paintings ugly, commenting on how art affects people generally, and such. This also does not mean that whenever people see something they aesthetically disagree with, they do not shrug their shoulders and cease to care. They may, for whatever reason, try to actively or passively oppose certain artworks, particularly if the art is public in nature and they cannot avoid the art. Subjective aesthetics also does not mean that trends can change over time, that art once considered bad and unpopular becomes popular and even considered the new "normal."
The parallels to morality are obvious. The reason it is more acceptable to say aesthetics are subjective is that, at least in the present time, not as much rides on our conceptions of what is good art. However, a large amount of our lives revolves our personal morality and the moral system that is accepted by the larger society. Most people can't act like morality is subjective and take a live-and-let-live conception of morality because no matter what, people have to live under moral systems. How this world views morality, what it permits and what it allows, directly affects what we care deeply about. Moral progress can simply be viewed as moral change with general trends based on prior moral views. It should be no surprise that we see the progression from the past to the present as progress; this is not due to some continued march towards moral perfection, but rather us seeing the past historical events that chronologically lead up to the current state of affairs. Moral reformers may start the trends we call moral progress, but some (I would venture most) of what we call unpopular moral reformers simply were ignored by the larger population.
To repeat what Terrapin Station stated, you are claiming believing nonobjective moral stances necessitates not caring about morality discussions anymore. It is like expecting all nihilists to cease to perform all actions because they think their life has no objective meaning.