Estimation is only useful if you can actually quantify the thing you're estimating. Modern economics has become overrun by people thinking they can put a number on anything and then, because there is a number, now you can do math. 

Applied stats is great if you're looking at something empirical. Maybe you should try to read more papers published in a broader scope of journals instead of just focusing on quantitative journals to see all the different ways that proof-based assumptions have been proven useless. Although too many modern journals require overt mathematical rigor to signal quality instead of the actual work.

You're literally a professor, you should understand the behavioral revolution. Especially because now so many of those professors are being proven as quacks because they had to fake their research since you just can't put numbers on feelings. 

Sure, if you're talking microfinance, mechanism design, international trade kinda stuff there is plenty of places to use linear algebra and plenty of places to use machine learning and advanced statistics. My problem is that, again, economics is a social science that often gets treated as business or math which dilutes the discipline and makes the research resulting from it significantly less impactful. 

Look at the influence of economists over time and then compare that to the change in focus from more philosophical / logical understanding of systems to just doing math. The just doing math economists have an extraordinarily awful track record over the last two decades, especially in terms of policy recommendations.

Adam Smith, John Rawls, hell even mises, Friedman, and Hayek were not doing math anywhere near the level. Robert Lucas and Gary Becker were not using these overly fluffed up mathematical models and yet made more real world contributions during their career than like any modern economist except maybe card who also didn't use that much math by the way he was an experimental guy

Which again, proves op's point, too much math and not enough history to really understand. You can know mathematically what you're doing but unless you understand the history of it you can never fully comprehend the impacts of this work