At least as far as I can tell, Haught simply doesn't understand Occam's razor or how people use it in their thinking. I say "As far as I can tell", because his explanation and exposition are so odd that I'm having trouble even making sense of them.
He accuses Dawkins and Hitchens of not understanding the idea, but what he accuses them of is bizarre. As far as I know, Dawkins and Hitchens never use the concept of Occam's razor to claim that there is only one level of explanation and that one level of explanation must exclude others. That is not an argument that I have seen - or at least have understand was being put - in any book that I've ever read by these writers. But that's what Haught thinks they are doing.
An example would be, "How do we explain the existence of the book God and the New Atheism by John Haught?"
Depending on our purposes, it might be salient to reply, "Westminster John Knox Press was sufficiently interested to publish it." Or it might be salient to reply, "Printers with equipment located in the US printed off copies, thus turning the proofed pages into an actual published book." Or we might talk about how John Haught wrote it. We could even talk about how Haught was inspired to write it by reading books by Dawkins, Hitchens, and Harris, among others. All of these claims are true, none of them are inconsistent with each other, and all of them give us an idea of the total causal process that took place. Each of them is supported by evidence. They are not competing claims.
So, if somebody reasoned, "God and the New Atheism came into existence as a result of John Haught's thoughts about the New Atheism; therefore, the book was not printed on equipment located in the US" that would, in fact, be fallacious. One possibly salient set of facts that forms part of the causal story does not necessarily exclude another set of facts.
I'm sure that Richard Dawkins and Christopher Hitchens understand this perfectly well.
So, is there no principle that can be called "Occam's Razor" or the "Principle of Parsimony"? Of course there is. But it's nothing to do with this sort of case. It's simply about using parsimony in our explanations if we want them to be more likely to be true. The idea was originally about not multiplying entities without necessity - though we really shouldn't get too hung up about entities. We really should be more concerned not to employ gratuitous assumptions.
Here's an example. I come home one night after a drink at the pub, only to find that a window has been smashed, the place has been ransacked, and many valuable items are missing.
Given a whole lot of background information that I have about such events, I may, quite reasonably, think that this was the work of a burglar.
This does not, however, exclude the possibility, "A burglar did it accompanied by an evil spirit." That's not how such arguments work. But the evil spirit's is not required for accounting for the evidence, while the possibility that there was no evil spirit involved makes it more likely that "A burglar did it accompanied by an evil spirit" is false than that "A burglar did it" is false. I am more likely to be correct if my explanation does not make use of this additional entity.
Note, however, that I may sometimes be more likely to be correct if I leave open the possibility of additional entities. Thus, I'm more likely to be correct if I say, "One or more burglars did it," than if I say "Exactly one burglar did it" (or, if it comes to that, "Exactly two burglars did it"). There may be practical reasons to test the exactly-one-burglar hypothesis and the exactly-two-burglars hypothesis separately, but the "one or more" way of stating the explanation is more likely to be correct. In any event, I need a reason if I am going to propose a precise number of burglars.
But what if I said, "It was done by one or more burglars who own horses"? Well, the horses are not necessary for the explanation. Perhaps it will turn out that the burglary was done by exactly two burglars, both of whom own horses. When we simply say, "By one or more burglars," we are not excluding that possibility. However, while the evidence gives us a reason to believe that there was at least one burglar, it doesn't give us any reason to believe anything about any horses. That might change if new evidence becomes available, such as a trail of horse hooves leading up to the smashed window. But on the information so far, we should only be talking about burglars. I have reason to believe in the existence of at least one burglar, but no reason to believe in any horses (or any evil spirits if it comes to that).
If we can give an explanation of some phenomenon in terms of "Naturalistic process X caused it", and if it doesn't assist us to believe that Naturalistic process X was helped along by an evil spirit, then we should not postulate the presence of the evil spirit. Again, that does not exclude the possibility of the evil spirit (though background knowledge suggesting a lack of evil spirits in the world may do so). But it renders the evil spirit claim gratuitous. Without some further evidence that could be plausibly explained by the auxiliary hypothesis of an evil spirit, we should not believe that an evil spirit played any role. (Indeed, if evidence turns up that can't be explained simply in terms of burglars, we might want to go down a different path entirely, rather than saving the burglar hypothesis by adding in an evil spirit ... but that's another issue.)
The same reasoning applies to God, or gods, as to an evil spirit. If we seem to have good prospects of explaining all the salient phenomena without recourse to these things, it doesn't exclude the possibility that they exist. But it gives us good reason not to postulate them in our explanations of the phenomena. Unless we have some independent reason to think they exist (for example if it turned out that some version of the ontological argument is sound) we should not posit that they do. It's pretty simple, really.