I think one of the main things that stuck with me throughout this paper was the assumption made about UG: namely, that whatever is in it is, by definition, arbitrary. I'll certainly grant that this is one way the knowledge of UG has been characterized sometimes, but I'm not sure I agree that it's a necessary property. So, if we find out that innate, language-specific biases have their origins in, say, pragmatic/communication biases, does that make them not UG? To my mind, it doesn't. But I think it does for the authors - and that seems to be one of their main claims for saying it couldn't possibly have evolved. So they strike out arbitrary, innate, domain-specific knowledge - does that means that all innate, domain-specific knowledge is ruled out? It seems like they want to claim that (and instead attribute everything to innate, domain-general processes), but I don't think I agree.
That being said, I do sympathize with the position that it seems more likely that language adapted to the brain, rather than the brain adapting to language. It seems reasonable to me that language evolution is too fast for the processes of gene evolution to catch up with, i.e., language is too much a "moving target".
A few things that also stuck with me:
p. 1135, where they say that aspects of language that are difficult to learn or process will be rapidly stamped out: Given this, should we expect that all persistent gaps (i.e., Russian inflectional paradigms, or why you can't say "Who did you see who did that?" to mean "Who did you see do that?" in English, even though you can do the equivalent in German) to be more difficult in some way? I suppose it's possible, but it seems less plausible to me.
p.1143: I love that they're looking at binding phenomena, because it's true that this is traditionally been an example held up by UG proponents as something that is very likely to be a part of UG. I'm not quite sure that their story about dependency resolution (how clauses get "closed off") would work for all environments where we use regular pronouns instead of reflexive ones, though. However, I think they're satisfied to show at least as few connections between these syntax principles and other non-syntax constraints - they say something to the effect of "This doesn't account for everything, but since no one can account for anything, this is as good as anything else." While it's true that no story accounts for everything yet, I suspect the syntactic accounts might go a bit further than the account the authors have sketched here.