Electronic – how to add a motor to a potentiometer

The controller you have identified will work.

However, bear in mind that this controller is NOT providing any isolation from the line voltage. This is risky enough with the control potentiometer shown -- its internal insulation needs to be rated for this type of application, and many inexpensive units are not. But if you eventually want to control this with an Arduino or other microcontroller, it will be up to you to provide the necessary galvanic isolation.

Given that you've said R7 is set to 0 ohms, the two fixed terminals of R6 are shorted together. When the wiper is at either extreme, its resistance to either fixed terminal will also be 0 ohms. When the resistance is 50%, both halves will be in parallel. Given that this appears to be a 50 k pot, we'd expect

$$R_{eq}=\frac{R_{max}}{2}||\frac{R_{max}}{2}=\frac{R_{max}}{4}=\frac{50000}{4}=12500$$

just as you measured.