Multimedia switch - audio - MIC bias
Analog headsets (typically using TRRS plugs, or two TRS plugs, or abusing a micro-USB connector like your headset) do need a bias voltage, so not getting any recorded signal might very well be caused by missing bias.
I can not find a data sheet for your USB multiplexer that deserves that name, just a product brief by Fairchild called "data sheet". The block diagram hints that the chip is likely able to pass the voltage from the MIC pin to the USB-plug headset at VBUS, but it does not say whether it needs any specific initialization for that. If you use that chip, you should have a more detailed data sheet that explains how a USB headset with microphone on VBUS is autodetected (if at all), and what you need to enable that. Furthermore, if you don't see a bias voltage at VUSB, I am confident that you also won't get sound input from VUSB back to the MIC pin.