Maybe Zoom works great for you, and it does all the things you want in a virtual background.
Nothing wrong with that, it’s what sold me on virtual backgrounds and shared virtual spaces in the first place. (I didn’t figure the rest of this out until I tried to recreate it in discord, because I had the darn green-screen set up!)
This still requires OBS Studio and the virtual camera plugin, but you do not need to set up the Face Cam scene with a camera source. In fact, if you have one, you probably have to either delete it or deactivate your webcam, so Zoom can use your webcam.
Launch Zoom. (If you’re doing this, I assume you have it configured the way you want already.) Enter a meeting and turn on your camera.
Create a new scene, “Zoom Background” in OBS.
Add a window capture source “Zoom Capture”. Choose your Zoom window from the window list.
Start Virtual Camera in OBS.
Launch discord (or whatever app), and point it to OBS Virtual Camera, and… voila. Literal zoom virtual backgrounds, but rebroadcast to discord.
(Subject to usual limitations, like limited meeting length in the free version, etc.)