by pretty much unpopular request

Really?
man, am I loath to re-enter this conversation, but

OK...
(meta: I can't wait to find out out whether this will qualify as not utterly stupid.)

First, friendly blogging advice? avoid putting links to long posts elsewhere at the top of your post. People with short attention spans click them, read them first, get bored, and may or may not get back to your OP.
The Mixing Memory guy has, imofwiw, some good points and some stupid points. *shrug*
I will point out that these two nearly juxtaposed sentences are contradictory:

While the female preference for "feminine" toys is obvious, the males don't seem to have a gender-preference at all.....In short, the data tells us zilch, zero, nada, nothing.

[and plus you know how I feel about "the data tells" but anyway]

But OK your laborious OP.

we now see hilarious reviews of the research like this one.

? That's not a review, it's another primary study on another species. Are you referring to their treatment of A&H02 in their Introduction? I read that, but I missed the hilarity. They are critical of A&H02 (making some of the same points as the Mixing Memory guy in fact).
(A critique, incidentally, to which Hines and Alexander have since replied; I don't know if perhaps any of your points are addressed there because I didn't read it.)

They are hypothesizing that the bases for alleged human sex differences in toy preference are evolved from a time before our hominid lineage began.

No they're not. SC, that's the Abstract, summarizing the whole paper. The sentence to which you refer begins "The results suggest that..." which (together with its positioning near the end) signifies that it's summarizing the Discussion (where some subjective interpretation and speculation is typically permitted), not the hypothesis. In fact, since the paper is available online, it's not hard to find the actual explicit hypothesis right there at the end of the Intro where it belongs:

To evaluate the possibility that sex differences in toy preferences can arise independent of these social and cognitive mechanisms, we therefore tested the hypothesis that vervet monkeys, like human beings, show sex differences in toy preferences.

---

and they believe their results show this. .

No, they believe their results "suggest" this. Do you think that's a trivial semantic niggle? I can assure you that in science it's not; it's an important distinction that is enforced by reviewers and editors.

Let’s take a look at the rationale for selecting these objects:

Let's! But to be fair, that rationale does not appear in the very first sentence of the Intro (which you have quoted), but rather in the Methods (2.1, 2nd 'graph). (Ah, you actually quote the passage below. So you knew.)

Time to play Spot the Spin Difference!
exhibit A:

These authors, though, hypothesize that human sex differences in toy preferences reflect evolved differences.

exhibit B:

The present research addressed the hypothesis that toy preferences may be associated with factors other than human social and cognitive development.

Let's play again!!
exhibit C:

To evaluate the possibility that sex differences in toy preferences can arise independent of these social and cognitive mechanisms, we therefore tested the hypothesis that vervet monkeys, like human beings, show sex differences in toy preferences.

exhibit D:

In other words, they expect that vervets will exhibit the same differences in preference that humans allegedly do, for the reason that the bases for these differences evolved in some ancestor, millions of years ago, common to both vervets and humans (and indeed, all primates).

Those are "other words" all right!
Do you (and strange gods) do this rhetorical shit on purpose, or do you really not notice? I'm sincerely asking this time.

they are not measuring play behavior, but merely contact with the objects.

Is that different from how preferences are quantified in human infants? (I don’t know the answer.)

But a study of sex differences in object preference for a given species should be designed with that species’ lifestyle – behaviors, including sex differences in activities; social structure - in mind.

It "should" be? But such a study would be addressing an entirely different question.

it is improper to generalize from a single culture in one moment to “boys” and “girls.”

I agree, but on the other hand it's not clear that the authors have done this. Are there cross-cultural human data available?

Again, the suggestion is that sex differences in preferences for features of these toys evolved early in primates.

Remember that they’re suggesting that a male preference for these objects evolved early in primates.

And again, remember that, a) no they never even suggest that a preference evolved in anybody specifically for these objects, and b) the actual suggestion that some sort of sex-specific preference behavior evolved before the common ancestor of Old World monkeys comes only later, after the results were in.
Of course I agree it's a weak stretch, but it's also explicitly extrapolative, never presented as a firm logical conclusion from the data presented. And if the data had turned out differently, they couldn't have said it even speculatively.
It's either dishonest or clueless of you to keep pretending that this is presented as the take-home message in the published article.

alleged girl preferences

I think the adjective you wanted there was 'empirical'

But it would be completely bizarre to argue that human females evolved a preference for modern objects currently associated with cooking as objects such that they would be drawn to their form without any knowledge of their function. To suggest that female vervets or our common ancestors, who obviously lack any knowledge of cooking, evolved a preference for these as objects is simply ludicrous. Species don’t evolve in anticipation of another species millions of years later using objects in a certain way.

I know! That would be bizarre! Ludicrous! Ridiculous!
Good thing that that's a strawfigure entirely of your own construction and that nobody--NOBODY!--not the authors, not me, not anybody else you ever may have argued with about this--has ever thought anything like that.

You know, SC, iirc this is the precise point on which our previous discussion foundered interminably. Your repeating it here makes me want to just say fuck it, because as I opined back then and then again more recently at Coyne's, I am forced to conclude that you just don't get it.

Plus, either you don't know how to read scientific literature or you are intentionally(?) deconstructing it for rhetorical purposes and either way it's overtaxing my eye-rollers.
I'm going to read the rest of your post but I don't intend to comment much more about it. This is seeming like a giant waste of everybody's time.

They conclude: .

Wrong. That is not a conclusion, it's explicitly a suggested implication of the data. What they actually concluded from the data is juuuust above there:

Children’s toys, therefore, appear to have differential value for males and females of at least two primate species, vervets and humans. This new finding provides
additional support for the hypothesis that sex differences in toy preferences can arise
independent of the social and cognitive mechanisms thought by many to be the primary
influences on toy preferences in human beings.

I'm going to point out that this far, and no farther, is as far as I have ever defended this article in any way. And it's exactly as far as I'm inclined to discuss it now.

but omg, SC. Next you actually quote the same passage I just quoted--the one that appears juuuuust above that other one?--but preface it with "They go on to argue that"?
That really looks intentional.

But as I’ve discussed, sex differences in these object “preferences” in vervets, should they appear, are likely to be a fluke, an artifact of a single study.

you're a nut. You haven't discussed this, you've simply asserted it. You're pulling your likelihoods right out of your ass.
But of course, likelihood aside, any single study in science may be a statistical fluke. Every scientist acknowledges this every day. We try hard to actually provide an accurate probability calculation for everything we conclude (implicitly tentatively) from our data. The probability is never 1.

In the conclusion they pull novel “explanations” out of the air.

“In the conclusion”? It's the fucking Discussion. They are obligated to attempt some sort of explanation for the data they are reporting. It's part of the whole formal convention of reporting science thing? And they don’t know. And so they're probably wrong. So what.

blah blah blah
and that’s the last I ever want to say or want to see on the subject, seriously.
It is Unimportant.