If the motor current draw was more than the unit could handle, smoke would happen.
Even if the unit didn't burn up there would be little likelihood that it would give any satisfactory results. The speed controls for dremmels are not rated for the current draw of larger tools like belt sanders. Router speed controllers are more robust, but as said, are for a different type of motor.
Motor speed in an AC motor is a function of the frequency of the power supply (hertz) and the way the motor is wound. If the speed is to be varied, something has to be changed in the equation. Since the motor usually has to stay the same in use, the power supply is what is changed. The voltage can be dropped, causing a loss in power, which is seen as a lower speed, but is terrible inefficient. The frequency can be changed, allowing most of the torque to be retained, but the external power control system to chance the frequency is not cheap.
So the two main solutions are a VFD ,to change the frequency, or a DC motor, which runs on variable voltage.
Believe me, if there was a simple, cheap, and efficient way to vary the speed of a motor above fractional HP, it would be on every shop tool in existence.
Stacy