I agree correlation is not causation. But the gradual improvement theory you posit is not in accord with the facts: the murder rate was high and increasing, assaults were at record levels, property crime was rampant, and then new policing approaches were introduced and crime fell. You can argue other factors were responsible. Levitt’s theory relating abortion rates to a delayed drop in crime is also a plausible mechanism supported by the data. But the drop was not gradual. Further the logic of statistical inference does not rule out Compstat as a causal mechanism- merely that the correlation is not dispositive.
I would make the additional argument that data-based decision making that promotes greater accountability has proven successful in a number of areas beyond policing. Predictive modeling is not a Delphic Oracle but, if properly applied, can be a useful tool. Despite its widespread adoption and success in numerous fields from sports team drafts to inventory management, you want to argue it somehow had nothing to do with the turnaround in NYC crime. What makes data useless in dealing with crime?