Matching unpopularity

It's not true that "no-one uses matching these days". It's a bit too definitive. For example, see Bilicka (2019) who recently published in the American Economic Review. I am sure it's possible to find other recent examples.

It's true however that the matching estimator is much less popular than the DiD or IV. This is because the results are not always consistent and robust across estimators. They sometimes differ depending on the estimator used (propensity score matching, PSM, or nearest neighbor matching, NNM) and the way in which the treatment and control groups are matched. It is also sometimes difficult to obtain matched samples that are as comparable as those in Bilicka's paper (see her Table 1).

As an example, there is a well know controversy in the Journal of Econometrics, involving Rajeev Dehejia, Jeffrey Smith and Petra Todd.

• Smith, Jeffrey and Petra Todd. 2005. "Does Matching Overcome LaLonde's Critique of Nonexperimental Estimators?" Journal of Econometrics 125(1-2):305-353.
• Dehejia response (2005)
• Smith, Jeffrey and Petra Todd. 2005. "Rejoinder." Journal of Econometrics 125(1-2):365-375.
• Dehejia Final Thoughts

My recommendations would be

1. to be very specific about how you match your treatment and control groups.
2. to provide lots of robustness checks to show that your results are not sensitive to a particular estimator, for example, PSM or NNM with Euclidean or Mahalanobis distances.

Good luck with your resubmission!