I'm not sure that you could point to any of those 'Darwinian' explanations as being neccessarily moral. After all there are proposed evolutionary explanations for behaviours such as murder, rape and genocide, and none of those is exactly a moral behaviour. Your example sounds like the sort of off the cuff evo-psych explanation which our member Holmes used to rail at. Without some reference and context it is hard to judge whether one should take such a specific suggestion as serious or not.
A much more likely evolutionary explanation of morality itself, rather than any specifc behaviour, would be the very one you yourself have gone for, the development of empathy and the ability to mentally put oneself into another's place.
Unless you think that your empathy derives from some spiritual medium then it is reasonable to identify it as a result of your mental processes and therefore of the structure of your brain resulting from evolution. So a darwinian explanation of actual morality per se is more likely to be based on the evolution of faculties which allow you to make the neccessary judgements of cause and effect and see them from the perspective of others than any particular ad hoc explanation of a specific behaviour.
There are indeed particular elements of the brain which seem to be associated with the act of putting ourselves in another's place or understanding their action's as similar to our own. A number of people have suggested that such elements, specifically mirror neurons, are the basis for 'theories of mind', an important element in empathy.
I see that in fact someone made a very similar argument to the one you quote in Kuresu's Why do right? thread and it does seem very ad hoc. I'm not sure if Kader was just putting that forward as the type of reason you might expect or actually claiming that it was the case, either way it certainly doesn't seem to be a well developed hypothesis.