Anyway, I think the author makes a bit of a straw man argument when saying that Dawkins' "moral Zeitgeist" doesn't determine a rational, consistent basis for morality. I don't believe that Dawkins was trying to show that.
I believe that Dawkins was trying to show that both religious and irreligious folks derive their morals from the same place for the most part: the moral Zeitgeist. This is a counter to the argument that religion is necessary for morality, because morality for most people is drawn from the Zeitgeist either way.
I haven't read Dawkins, so I may be wrong in my assertions of what Dawkins intended, but it feels like the author covertly (or accidentally) tried to shift the focus away from countering the "religion is necessary for morals" argument, onto a different topic (that of defining rational morals).