But the interventions that they consider are almost all lockdown-type (restrictions on motion), and their methodology assigns all the credit to the interventions considered. So any benefit from effects they didn't consider--mask use, voluntary changes in behavior, earlier herd immunity due to heterogeneity, etc.--gets falsely assigned to the lockdowns. So at best, this study may show the relative value of different lockdown-type interventions, for example their conclusion that school closures were relatively ineffective. It absolutely cannot show that the lockdowns were the reason for the difference between reality and that first-order model--that's an input assumption, not a conclusion.
Even the interventions considered show surprising results. For example, "WFH, no gathering, other social dist." in Italy shows a positive effect on the growth rate. I don't think that means working from home makes the virus spread faster; rather, I think they're mostly curve-fitting to noise, even by the already doubtful standards of econometrics.
I understand scientists' impulse to justify the lockdowns in retrospect, given the high social and financial cost (and I agree that in many cases that cost was justified!--New York and Lombardy certainly wish they'd closed sooner). I'm disturbed that the authors, Nature, and the popular press all seem happy to disregard the truth in order to provide that justification, and I fear the resulting public loss of confidence in science will cause harm for many years to come.
Duh lock down slows spread, nothing interesting about that conclusion. What we need is broad analyses of the non-covid (mental) health impacts, economics, and a lot more. Decisions were made with lack of information and we can do better now.