OBS Studio. It's very powerful, classic open source, with a not amazing UI, but, it does everything mmhmm does and a lot more. I use both every day.

I tried OBS studio, but could not figure out how to use it properly. Ended up using Xsplit Vcam instead, which is really good.

I think I managed to remove the background in a really, really crud way with OBS studio. Like using a chroma key (green screen). But Xplit does it with any noisy apartment background.

Xsplit offers an integration with OBS, too.

https://www.xsplit.com/partners/obs

I just found this which may fit the bill: https://github.com/fangfufu/Linux-Fake-Background-Webcam

I'm thinking of making something like that, the hard part if working with QuickTime apis on Mac to get a virtual camera going.