Pereltsvaig and Lewis begin with the historiographic background, outlining the politics and ideologies of debates over Indo-European down to the 1970s, and then describing the specific Anatolia vs Steppes debate.
They then explain four kinds of problems with the model used in the paper. Linguistically, it fails to distinguish innovations from retentions, or to detect borrowings, and takes a purely lexical approach.
"The most glaring blunder concerns Romani: not only is Romani mistakenly taken to have split off from the rest of the Indo-Aryan branch before any other language, the date of the split is wrong as well. Bouckaert et al. place it around 3,500 years ago; linguistic evidence suggests a much later date around 1,000 years ago."
When it comes to dating language branches, the model's Bayesian methods offer "only a slight improvement over the older and thoroughly discredited approach of glottochronology". And the model produces other historical and geographical predictions that are clearly wrong. The model also makes unwarranted assumptions, notably that language structure and spread is similar enough to biological evolution for biological methods to be used.
"The model erroneously assumes that languages are universally patterned into spatially discrete, non-overlapping units"
"in the Gray-Atkinson scheme, not only do new languages emerge strictly on the basis of evolutionary descent from a single common ancestor, but they also do so only on the basis of changes in words"
Pereltsvaig and Lewis then look at some of the things they think historical linguistics can do. They touch on relative dating and the constraints on PIE, linguistic evidence for the location of the IE homeland, and the social processes involved in the linguistic outcomes of migrations, notably relationships between pastoral and settled populations. And they make some general observations on historical linguistics, looking at alternatives to words as "atoms" of language that can be used in phylogenetic tree reconstructions.
A conclusion considers broader epistemological issues. Here the Bouckaert paper is described as an "intellectual disorder", with the misuse of mathematics an example of "extreme rationalism"; this is compared with the damage done to geography by physics envy and "the quantitative revolution". (To me it seems an example of a "drive-by regression", where the rigour of a mathematical tool, in this case Bayesian phylogenetics, is used to obscure proper consideration of how well it actually works.) As epistemology goes The Indo-European Controversy is not sophisticated — it falls back, for example, on vaguely arguing that "a lot of evidence" points towards the Revised Steppe Hypothesis — but historians and philosophers of science are likely to find it useful as a case study.
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