Did a little further research on my own, since this question had been a little slow to elicit any other proposed answers.
The vintage Sunbeam Mixmaster models from the mid-20th century through the 1970s (including models like the 1-7A, 1-8A/1-8B, and Model 12) use a variable resistor-based speed control mechanism to control and vary the beater motor/speed. The mechanical means by which the speed dial controls the resistance in the motor circuit results in a variable speed.
The design is a mechanical way to operate a rheostat (a two-terminal variable resistor) or a potentiometer (a three-terminal variable resistor used as a rheostat) that is connected in series with the motor.
The Beveled, Circular Track (The Resistor) inside the rotating cap is a circular track that is the actual resistance element. It's typically made of a non-conductive ceramic or phenolic material coated with a thin layer of resistive material (like carbon or a coiled wire). The amount of resistance along this track is what controls the current flow.
The Rotating Actuator/Pin is part of the rotating dial on the outside of the mixer turns a shaft that moves a small metal piece, which is often shaped like a pin or brush. This pin/brush is called the wiper or contact arm.
As you turn the dial, the wiper moves along the circular resistive track and the wiper acts like a short circuit, tapping the electrical current at different points along the track. At low speed when the dial is set to a low speed, the wiper is positioned so that the current has to travel through the maximum length of the resistive material before reaching the motor. High resistance restricts the current, making the motor turn slowly.
When the dial is turned to a high speed, the wiper moves to a point on the track where the current has to travel through the minimum length of the resistive material (or sometimes directly to a contact that bypasses the resistor entirely). Low resistance allows maximum current to reach the motor for high speed.
I'm able to visualize this configuration, and I'm not entirely sure this exactly the setup that I have in my Mixmaster model. While I've tested my resistor and it tested fine - I still think it is the problem, because while the mixer will engage the beater motors at the lowest speed setting, the beater motor speed seems to be at a consistently high RPM - suggesting the contact is either bypassing the resistor, or the resistor is registering very low resistance and the motor is always running at its maximum speed/RPM.
Any other insights or suggestions, based on the clues and upstream motor functionality summary?
