Structural estimation is useful, not only for the estimates (whose credibility is often undercut by the panoply of supporting assumptions that are required to obtain them), but for understanding the empirical predictions of the theory.
Do we do this with DSGEs? We try to replicate known patterns in the data, but I don’t think we use these models to find potential new patterns.
In one of my myriad of yet-unfinished papers I report things I call “simulated comparative statics”. I can’t solve the model, so I have the computer generate numerical derivatives. People don’t like ’em. Can I market them better? “Simulated theorems”? “Generated predictions”? “Computed hypotheses”?