This is incorrect. These fans use PWM control. The power is provided directly to the fan via a separate wire. The fan assembly includes a speed controller. The PWM signal is used to tell that controller how fast to run the fan. The PWM signal itself is logic-level, it does not actively drive anything at all. It can source very little current. Which is good, because the PWM input to the fan will have high impedance and very little current can flow in to it anyway. The fan’s built in controller does all the work including switching the voltage from the separate 12V input to the motor windings to feed power to the motor within.
These fans are 4 wire. Power, Ground, PWM speed input (logic level) and Tachometer speed output (logic level).
What you describe is how DC brushed motor speed controls work. These fans, however, are Brushless motors. They have to have a controller built in in order to work.
It’s every bit as simple as it sounds. GF has no interest, I suspect, because the design doesn’t contain the features that are necessary to provide a switched receptacle for external fan control, not the least of which would be a socket to plug the external load in to.
If the switch to my external fan wasn’t literally only 2 feet away from the button on the GF, I’d have made something to do this already. Starting from scratch, I would bet it’s only a few hours work.
For those less inclined to roll their own, something like this will do the job and requires only the most basic wiring knowledge.
https://www.amazon.com/Current-Sensing-Normally-Adjustable-SZC23-NO-AL-CH/dp/B07N1P6TWL
The AC in to the GF passes through this device. It has a settable threshold for the “trip level”. When the current draw exceeds the trip point, the switched side of the device is activated. The GF unquestionably draws more current when running than when idle. That’s all that’s necessary for this gizmo to function.