mmm, can't change physics,lol
Brushless AC is an expensive way to go, it has power, but the cost would be through the roof. These are normally kept in the domain of high end industrial such as Robotics and product handling equipment.
AC induction, low cost, low tech!!
Used for fans and constant speed,constant torque applications.
Brushless DC (or servo) used to be the domain of high accuracy applications, but AC servos, are a very real options these days.
Brushed DC, series or shunt wound(in tools it's usually series for safety reasons)
Cheap, powerful, easy to control and maintain, why re invent the wheel?
I fly RC helicopters for fun, they use Brushless high RPM DC motors of pretty low Wattage. A router with the same technology would cost $1000s and would offer nothing new.