I share your rejection of moral relativism, at least in any absolute or objective sense of the term. I would argue that framing the choice between absolute objective morality and absolute moral relativism creates the impression that we are left with a false choice between these two extremes. I also agree with you that the cultural defense for certain practices is unworkable. Child abuse or the oppression of women, for example, does really produce particular outcomes both for the individuals involved and the societies that tolerate or promote these behaviours. These outcomes can be objectively verified.
Once we’ve verified these outcomes, the question of right or wrong is reduced to what our own personal subjective reaction is to that behaviour and its outcome. For example, how would we feel if we were forced to wear a burka, denied the right to vote and threatened with stoning if we tried to drive a car or receive an education? Given we all, with too few exceptions to bother with here, all feel a particular way about being treated that way it’s safe to conclude those actually living under those conditions would almost certainly rather not live under them either if allowed to make an informed choice. That they may be in some cases too ignorant of the outside world to know there’s an alternative does not change the objective truth that physical and emotional abuse nonetheless has very real physiological and psychological impacts upon those experiencing it. The fact that they must be kept in ignorance to maintain the cultural practice only compounds the immorality further. Who among us would choose ignorance of a better alternative to our current situation rather than knowledge of it, and then insist upon the right to choose between those alternatives to the greatest degree possible?
Objective morality relies upon our own subjective experience of either of the immorality directly as a victim or indirectly as a witness to it. Objectivity and subjectivity, at least in a social context, are just two sides of the same coin. But we shouldn’t confuse subjective experience with moral relativity, at least not if by that we mean we can reach no moral conclusions about anything at all. In some cases we may be the only person in the room that feels a certain way about something but acts like rape, murder, and genocide usually are not among them. Obviously, when it comes down to the morality of what somebody said or certain acts of civil disobedience there’s frequently more room for interpretation and debate. But as a rule the more nuanced the case the less likely it is the situation involves serious physical and/or psychological consequences for those involved let alone a more or less universal reaction among those witnessing it. If I call someone a jerk for butting in line at the movie theatre many, perhaps even most people, will say he deserved it. However, if I pull out a revolver and shoot him for it virtually everyone will say he didn’t. Relativism, in this case, has everything to do with the relative impact of my action in response to a relatively small moral offense on his part and nothing to do with the absence of morality entirely.