I found a workable way around this problem -- in the Sound Control Panel for the Model 12 recording device on the Advanced tab, if I change the Default Format from "2 channel, 24 bit, 48000 Hz (Studio Quality)" to "12 channel, 24 bit, 48000 Hz (Studio Quality)" it starts working immediately. Whatever is going to the Stereo Mix on the Model 12 shows up on in the Sound Control Panel meters and applications like Zoom, Teams and Discord can use the Model 12 as an input. Oddly enough, Voice Recorder stops working -- it fails with "Your microphone is not working" but I can live with that. Note that the option for 12 channels is not available until after installing the drivers from TASCAM.
FYI, I also sent the following to TASCAM support but have not heard back from them:
The Model 12 shows up as both a input (mic) and output (speaker) devices when connected via USB. The speaker function works as expected-audio from Windows can be routed to channels 1&2 on the M12, however the mic function does not. Audio from the M12 is not sent back to Windows, even in stereo mix mode to outputs 1&2. Windows apps like Voice Recorder, Zoom or sound control panel do not see any input from the M12. Same behavior either before or after installing the TASCAM drivers.
ASIO works as expected and interestingly, launching a application using the ASIO interface like Reaper causes the Windows input to start working as long as Reaper is in the foreground. Reaper is minimized or I switch to another window, the mic stops working, then works again if Reapers is back in the foreground. Arming any track on in Reaper or playing any audio, even silence, to the M12, the mic input in Windows also continues to work.
It's almost as if the USB output of the M12 goes to sleep and stops communicating to the Windows audio interface and having any active ASIO application wakes it up. The behavior is very repeatable and I have reproduced it on a total of 4 different Windows 10 computers ranging from 1909 to 20H2 to 21H2. One of which has other audio applications on it, another is a clean system with just bare-bones Windows 10.
In comparison, my iXR works perfectly after installing its drivers on a computer -- both input and output to/from Windows applications works without having to launch any ASIO applications to get it to start sending audio back to Windows.
The M12 came with firmware 1.22 and I even tried reinstalling 1.22 to make sure it was correctly installed, but that made no difference. It does not seem like I can downgrade the firmware to 1.21. In Device Manager, it shows the TEAC Corp driver 1/10/2020 15.43.39.95 under the Sound, video and game controllers section and the Microsoft driver 12/6/2019 10.0.19041.1 in the Audio inputs and output section.