It's not an appeal to artificial authority. It's a very likely root cause and comes with a solution even: get a different data set or adjust your existing data set. Your response rings of No true Scotsman to me since you can argue any analysis is not rigorous enough or doesn't cover all potential issues.
edit: And a statistical analysis isn't some sort of magic data genie. Statistics can give rigorous results because it makes strong assumptions. If those assumptions don't hold then the results aren't rigorous anymore. A trillion parameters model can pull interactions out of your data that almost no statistical analysis of the data would identify ahead of time. So what you need to analyze is the model and try to infer why it's predicting different certain results and then work backwards from there.
