So I emailed Professor Putnam and asked him if he agreed with their use of his work. He replied:
I have a mixed view about the Spirit Level. On the one hand, I believe that inequality is bad for society in many ways, just as that book argues. On the other hand, Pickett and Williamson’s work has been heavily (and I believe correctly) criticized as methodologically flawed. (For example, they don’t really show that the relationship between inequality and other bad things is causal, though they assume it is.) I hope that they (or others) will pursue that basic hypothesis in ways that are more scientifically persuasive.
So neither support nor condemnation for the work. This is similar to my own sense that much of their argument about inequality being harmful feels intuitively correct, but that their numbers and examples are often questionable.