There's far more than just a correlation here, but the fact that the correlation holds for counties, states/provinces, and at the country level is a good indicator that the correlation is not spurious. There is also well established biology. And lastly, even the lag is appropriate. It's not like they picked some number at random for the lag. The lag is important, because the lag comes from the difference between critical stages of brain development and the time to peak risk for criminal behaviour. That is well supported again, by biology, and by sociology/criminology. The last bit, is that the correlation and variability explained by the relationship isn't just to one indicator. For crime, it's against many different types of crime. Assault, robbery, murder, rape.
Of course it's not certain, but that is a strong case. The correlation suggests association. The biology provides a causal mechanism. The sociology and biology suggest time dependencies. None of that is easily dismissed.
Still I can think of a criticism, it would be nice to see a negative control where lead values haven't followed a similar trend to those geographies that were studied. And for those jurisdictions where the association isn't significant, it would be nice to see how the regressions perform with and without other leading associations in those areas.