Shh, you've let the cat out of the bag- sure they are essentially synchronous motors but Brushless DC has been the name used (I guess because they operate from a DC source) After all, one could consider that a conventional DC motor is a mechanically switched synchronous machine with its field stationary and there used to horrible machines which could run on DC and produce AC or vice versa (not MG sets but with a common stator winding).
Agreed as to the eventual demise of the single phase AC motors - As of now many such as in corded lawnmowers and tools have already been replaced.