Would large, low aspect ratio fan/propeller blades be more efficient? Why not?
In the case of propellers, there are a number of operating and design considerations which must be taken into account here.
For example, while it is true that imparting a small impulse to a large mass wastes less energy than adding a large impulse to a small mass using a propeller, there are practical limits to how big you can make the propeller diameter before the ground handling of the airplane becomes unmanageably awkward. A huge prop might be more efficient, but you'd need a boarding ladder to get into the cockpit behind that huge prop and the landing gear struts would need to be longer and stiffer and hence heavier. Oh yes- and the prop itself would be significantly heavier.
In addition, to drive a big prop slowly and still develop full power with a piston engine requires a reduction gear of some sort between the 2400RPM of the crankshaft and whatever the turning speed of the slow prop was. The gearbox adds weight and requires an overhaul when its time is run out, and that adds cost- and a potential failure mode that is absent in direct-drive designs.
Finally, gearing down the engine to turn a slow prop magnifies the transmitted torque to the prop and therefore from the prop to the engine and airframe. That reaction torque has to be countered somehow or else opening the throttle will not just turn the prop but also roll the aircraft backwards around the crankshaft axis.
100 years of experience with geared and direct drive piston engines and all sorts of props has yielded the best solution in the form of a direct-drive engine spinning a variable pitch prop with a diameter that puts the blade tips comfortably subsonic at the max power setting of the engine.