Their general thesis seems to be that these societies were highly conscious about the political and hierarchical implications of their modes of production/ positions of authority … you name it, and thus consciously decided to form their societies in a way that accommodated their preferred type of social relations.

I don't think this is the main theses of the book, although they do infer this at times. I think their main goal is to simply dispel myths surrounding early man: that they were all innocent, or all violent, or all stupid; or that certain ways of organising societies were natural or inevitable, or that human societies all developed in the same linear way. By helping people to gain a new understanding of the past, I think they're trying to help people think differently about how things could be in the future. That, for example, a city doesn't necessarily have to be hierarchical and authoritarian in order for it to function. There's also the stuff about finding out how we got "stuck" but I'm not really sure where they're going with that since they've only touched on it briefly where I am in the book.

That said, I think that in the places where they do argue that some societies were arranged in a conscious way isn't necessarily incorrect per se. Like its possible that some societies had an aristocracy at some point but ended up overthrowing them and arranging things with the intention of stopping it from happening again - this wouldn't be too far-fetched if we're to believe earlier humans were as intelligent as we are, why wouldn't some of them seek to overthrow authoritarian systems just as we do, or punish those who seek to place themselves over others in the first place? Their thoughts are shaped by their conditions and experiences just as ours are, so I don't think its too unreasonable to say that some of these more egalitarian societies were responses to or attempts to prevent more hierarchical ones. Although, even if that was the case, I agree that the Davids would need to do a much better job in proving it, if something like that could even be proved at all.

And with this framing, I don’t really see how this: [...] is supported by this [...]

Yeah I agree with this. From what I can tell, their source doesn't exactly back up what they say, and I don't like that they don't mention the climate factors at play. Its still possible that they're partly right - that more people could have kept farming using the more resilient crops but perhaps chose not to because the crops failures made them lose trust in the benefits of agriculture, so they turned back to foraging the plentiful supply of wild hazelnuts instead. This could also explain the several centuries without agriculture even with the favourable climate restored. But again, they need to be more explicit with their reasoning for thinking it was a conscious decision and they need to provide better evidence for this.