I finally figured it out, but it needs a bit of “wrapping your head around things”.
When you set the USB option for a channel’s input to POST what it does is that the k-mix will no longer send the audio that goes into input N to the computer, instead it will send what goes out from output N.
So to be able to record stuff post-fader and post-fx you need to set the k-mix up so that each input channel is mapped to an output channel (i.e. in 1 and 2 to main, 2 and 3 to AUX 1 etc.) then flip all the switches in the USB page of the editor to POST.
Not the most intuitive way to handle it, but fortunately the k-mix has presets, and you can devote one just to this. You do have to renounce to using AUX busses for this though.
Not really sure what the thinking behind this setting is.
And something else I’ve noticed while messing around with the k-mix’s channels 1 and 2: they are pretty unusable with anything that has a normal line level and no on-board volume control, since you’ll always end up overdriving the inputs.