Here is an example of a large turboprop aircraft designed for efficient high-speed (>380 kts) cruise. The audio should give you some idea of why this particular approach is not more widely used.
While this aircraft proved that you can actually operate a propeller at supersonic conditions, there was a good reason it was nicknamed the "Thunderscreech".
While the theoretical comparisons between props/un-ducted fans/turbofans is interesting, in practice most commercial aircraft will use turbofans for the foreseeable future. One only needs to look at the recent turbofan development work being done at P&W, GE and R-R. It is very impressive and is progressing at a far more rapid pace than propellers and propfans.
Consider
R-R's UltraFan program, which they plan to put in service within a decade. It will have variable pitch fan blades, a reduction gearbox driving the very large diameter fan at optimum speed, a BR >15:1, and an overall
PR >70:1.
GE is currently testing subsystems for their
adaptive cycle engine, which will significantly improve efficiency of turbofan engines used on very high-speed aircraft.