The absence of new empirical sources for mathematics seems to me more a consequence of the stagnation of theoretical physics than the social structure of the mathematical community.

From my perspective in applied math things are exactly opposite. We're practically drowning in empirical sources begging for better numerics. There is hardly a scientific or engineering field today that couldn't be revolutionized by a breakthrough in a relevant area currently under study in the applied math community. The problem isn't lack of necessity, inspiration, or motivation, the problem is the problems are damn hard, and I'm far from convinced that we attract the best minds.

Even with very clear, objective goals, progress can still be painfully slow!

In my own view insufficient emphasis on exposition has played a significant role in whatever stagnation has occurred within the mathematical community since, e.g. the 1800's.

I very much agree. Incentives are out of line for most areas of science, but I think the fallout from this is especially poignant for math. I view part of our discipline's responsibility as curating the common mathematical knowledge used by all other disciplines.