Why AC is not used to excite synchronous machines?
A synchronous motor cannot have an AC supply on the rotor as the rotor is a single-phase element designed just to provide magnetic flux.  Since the stator is driven from a three phase source, you would need to provide a three phase, but OUT OF PHASE supply to the rotor in order to generate and maintain magnetic flux.  Tough to provide with two wires.  For low power applications, synchronous motors often have simple permanent magnets in the rotor and do not need an external supply.  

High power synchronous motors can be used to control the power-factor presented back to the line, but there are usually better options for power-factor management.

What you are describing isn't a synchronous motor, but a wound-rotor motor.  Wound-rotor motors have a three phase wound rotor (hence the name).  External resistance is switched into the rotor windings to inefficiently control speed, and change the startup loading characteristics of the motor.