I use OBS Studio with OBS-VirtualCam [0] to attend virtual lectures & hold meetings for my team. I've found it to be incredibly convenient because you can control nearly everything with scenes and the audio controls.
Before meetings start, I can broadcast music and display announcements, and then without having to hit a jarring "End Screenshare" can switch to my webcam and start a meeting. Live demos and presentations are another scene with the desktop/window/browser and webcam. 100% would recommend.
I'm doing something similar on Linux, although a bit more complicated.
I'm using v4l2loopback [0] to create a dummy video device, ffmpeg to create a stream endpoint that streams into the dummy video device, then setting up OBS to stream to localhost.
It is actually really nice to have the capability to fully control what is going in to the video input.
I haven't run into a need to also change the audio input yet but if it becomes necessary, it should be possible to set up loopback with ALSA.
Could you detail on how you are creating the ffmpeg streaming endpoint? That would be very heplful!