Creating a custom playback device
The simplest way to achieve what you want is to connect your potentiometers to something which sends their position to mixer software running on the PC, mix the signals inside the PC and then use one USB (or builtin) sound-card for output.

You can either make the potentiometer to usb device yourself or buy something like this:



(Image source: Korg)

(This is a Korg "nanoKONTROL2", many other similar devices exist)

Computer music people call this sort of device a "control surface". It has knobs and sliders which can be set to control all sorts of things (gain/volume, balance, reverb, which instrument is playing etc.). Lots of software exists to use them as mixers, indeed that's one of the uses for which they are designed.

If you wanted to do exactly what you describe you could take S/PDIF outputs from the usb sound cards, mix them in an FPGA weighted by the data from your pots, and then output through a DAC. If you've not done much electronics before I wouldn't recommend an HDL S/PDIF receiver as a first project.