Pinker is just TOO disingenuous.... IMHO FWIW.
IMHO, Pinker leaves himself no basis whatever by which to criticize anything. Indeed, Pinker's entire point seems to be that, once we understand man as the machine he putatively is (according to Pinker's grotesquely reductionist view), then questions about the "ghost in the machine" (does he ever source this quip to Julian Huxley, its author?) become moot. Irrelevant.
Of course, he knows as well as you or I do that whatever moral vision man has ever expressed down the ages, he never entrusted its communication to the language of a "machine" as his mouthpiece -- unless he needed an insanity defense....
So, what do you suppose this guy is really up to?
Pinker may be one of these scientific propagandists who make great claims for the latest research of science and its applicability to life. Iconoclasm, overconfidence, arrogance and prickliness, are common traits of this sort of person. When a new idea or scientific finding comes along it's seen as the big answer to everything. After the dust clears, we may find out that Pinker was right about some things and wildly wrong about others.
You've already noted the underlying materialist assumption of this whole discussion. In a Darwinian sense, this ends up giving us a situation in which the basic moral foundation is "whatever works." And, contrary to what the review, at least, implies, there is no particular reason that the utopian view cannot account for inherited differences -- if only as something to be actively extinguished.
If there's an elephant in the room in this review (I've not even seen the cover of the book, so I can't comment on it...), it's the apparent absence of Pink's having conducted any meaningful discussion of right and wrong. For example, we see discussion of the tragic and utopian views, but there doesn't seem to be anything for them to be tragic or utopian about.
(And, of course, one must also consider the ramifications of the idea that in Pink's conception, the "utopian view" would in some degree have to be inherited....)
IMHO, when you get right down to it, he's doing what a lot of other people do: he's either trying to justify the last 6 Commandments without having to invoke those pesky first four; or he's setting up an argument for some variation on the idea of might makes right.