@Wynvu this won't work. Exactly the setup I had in mind. But learned Fuji didn't start supporting HDMI live feed out on the X100 until the X100F.
I thought about selling my X100T and levelling up to the X100F. But even on eBay, that was gonna cost £300. So when I found an XF10 for £200 I jumped on it. I find it basically works for streaming as I’d hoped, except there is no way to remove the onscreen touch control UI, so I end up cropping my image dimension down in OBS software to send a clean image to Zoom / Meet / whatever.
Note I use a much cheaper generic HDMI to USB converter than the Elgato and haven't had any issues.