Huemer's views belong to what is called moral intuitionism, we have moral intuitions that track objective values, contra Mackie's argument from queerness that such intuitions would be unprecedented, and our having them an unmotivated leap. Huemer's argument from history is a response to what is called evolutionary debunking arguments against moral realism, it is modeled on the traditional historical argument for the existence of God, which worked better when the plurality of religions was less known. Modern versions identify what is common to most current religions, which includes much of the morality. Huemer just keeps the morality, which allows to be even more inclusive, e.g. of secular views.
The prevailing modern idea is that history would give evidence, if any, of adaptive utility of moral norms for social groups, rather than of objective moral values. Such utility is pretty commonly acknowledged. Lorenz famously traced the origins of morality to animal evolution, evolutionary ethics is built around the concept, Mackie in Ethics explains deontology as a device to avoid inferior outcomes in prisoner's dilemma situations. The debunking arguments go further in comcluding that even if objective moral values existed we are not likely to get to know them, because our moral sensibilities are shaped by adaptive factors. Here is Huemer in A liberal realist answer to debunking skeptics: the empirical case for realism:
"Debunking skeptics claim that our moral beliefs are formed by processes unsuited to identifying objective facts, such as emotions inculcated by our genes and culture; therefore, they say, even if there are objective moral facts, we probably don’t know them. I argue that the debunking skeptics cannot explain the pervasive trend toward liberalization of values over human history, and that the best explanation is the realist’s: humanity is becoming increasingly liberal because liberalism is the objectively correct moral stance."
In a sense, the debunking is an adaptation of Plantinga's evolutionary argument against naturalism, so the responses to it can, in turn, be adapted by moral realists. The problem is with the disanalogy between ethics and natural science related to the aforementioned queerness, the values/norms are not of the nature of facts, they are imperatives rather than declaratives. And one can explain the liberalization and convergence of moral norms by far more mundane utilitarian means. Morton reviews arguments against Huemer type views in A New Evolutionary Debunking Argument Against Moral Realism, and Hopster in Explaining historical moral convergence. See also Evolutionary Debunking, Moral Realism and Moral Knowledge by Shafer-Landau, who is more sympathetic to moral realism:
"The argument claims that given the extent of evolutionary influence on our moral faculties, and assuming the truth of moral realism, it would be a massive coincidence were our moral faculties reliable ones. Given this coincidence, any presumptive warrant enjoyed by our moral beliefs is defeated. So if moral realism is true, then we can have no warranted moral beliefs, and hence no moral knowledge. In response, I first develop what is perhaps the most natural reply on behalf of realism – namely, that many of our highly presumptively warranted moral beliefs are immune to evolutionary influence and so can be used to assess and eventually resuscitate the epistemic merits of those that have been subject to such influence. I then identify five distinct ways in which the charge of massive coincidence has been understood and defended. I argue that each interpretation is subject to serious worries. If I am right, these putative defeaters are themselves subject to defeat. Thus many of our moral beliefs continue to be highly warranted, even if moral realism is true."
