But let's for a minute assume that the core of the fanjet contributes very little to the forward motion and assume that the fanjet produces most of the thrust (which I think I've heard is the case).
It is the gas generator that produces all the energy by continuous thermal reaction. Most of the thrust is produced by converting this energy to rotational force to drive the fan; some thrust is produced by the jet efflux from the gas generator. The difference between the two is the bypass ratio.
Now, if that's a "sweet spot" and an efficient way of reaching high cruise speeds and thrust envelopes it begs the inevitable questions: why not use ducted fans on piston and turboprop engine aircraft then? Why is the propeller still around if the fanjet is such a fantastic thing of efficiency and so good at reaching higher speeds?
Because they have different range, speed and payload requirements. The propeller is more efficient at lower a/c speeds. The advantages are debated in Keesje's recent thread proposing a new short range turboprop.The faster contra-rotating exceptions mentioned by chornedsnorkack are too noisy and do not meet the current sound pollution regulations.