One thing that has always struck me as rather odd about the B737 is the way the main wheels still remain exposed even after retraction. Presumably there must be at least some aerodynamic penalty for that, so does anyone know if Boeing ever considered “fixing” it?

The aerodynamics of the B737 are “dirty”, especially in that area. You have quite a thick boundary layer, due to many influences (antennas, air conditions, heat exchanger). That is the downside.
Advantage is less structural weight, less complexity, better cooling for the brakes (remember: the B737 comes without brake fan, big issue for some operators).