The debate ended in 2005.
Shortly after this, Chomsky (2005/2008 (written in 2005, and circulated, published in 2008) wrote On Phases which did not acknowledge anything from his previous papers co-written by Hauser and Fitch.
In Chomsky (2005/2008) he proposes this (p. 5 in the 2005 version, and I'd assume the 5th page of the 2008 version):
Suppose that some ancestor, perhaps about 60,000 years ago, underwent a slight mutation rewiring the brain, yielding unbounded Merge. Then he or she would at once have had available an infinite array of structured expressions for use in thought (planning, interpretation, etc.), gaining selectional advantages transmitted to offspring, capacities that came to dominate, yielding the dramatic and rather sudden changes found in the archeological record. Speculation, of course, as are all such stories, but about the simplest one imaginable, consistent with what little is known, and presupposed in some form (often tacitly) in all speculations about the matter. Since the integration of language precursors into FL together with innovation of unbounded Merge would have been sudden (in evolutionary time), effects of path-dependent evolution and other complexities that underlie the logic of Jacobian “bricolage” might be secondary phenomena, and evolution to a form of FL approaching SMT, not too surprising.
There is nothing about spandrel structures, or anything else. It is nothing but pure speculation. He does not even cite himself from Chomsky, Hauser & Fitch.
Two years after On Phases was published, one of Chomsky's co-authors and co-workers Marc Hauser was embattled in a massive scandal over scientific misconduct, and resigned. Accusations of scientific misconduct arose four years before his resignation in 2011.
Hauser has gone on to continue publishing works in language origin, but is unaffiliated with any universities.
I do not know what Chomsky's thoughts are on the Spandrel Structure idea for the biology of Language Origin. He has not written about it (I reckon) since 2005.
Chomsky's work gravitated into biolinguistics with the likes of Samuel Epstein, around 2005 and hasn't consulted with biology since then, to my knowledge.
His work towards the end of 2005 no longer seems to acknowledge the things hypothesised in Chomsky, Hauser, & Fitch.
So, it terminated right where you stopped reading, to answer the question explicitly. You got to the end of it. Afterwards, Chomsky went in a different direction.
Back to Hauser--in 2014 Hauser et al., published a paper in Frontiers about the evolution of language (interestingly, Hauser affiliates himself as a CEO of Risk Eraser in the author affiliations). The paper does not conclude much and is more of Squib if you ask me. It provides suggestions for future research, and summarises the problems so far.
Chomsky, N. (2008). On phases. Current Studies in Linguistics Series, 45, 133.
The 2005 version is available at:
Hauser, M. D., Yang, C., Berwick, R. C., Tattersall, I., Ryan, M. J., Watumull, J., ... & Lewontin, R. C. (2014). The mystery of language evolution. Frontiers in psychology, 5, 401.