no, not missing anything... just the challenge of using a tool designed for interactive compositing to capture an entire display of a different application(s) on a single monitor
I use dual monitor, even when I started with a laptop (due to my use case).
There are plenty who use OBS Studio (and other similar compositing apps) on a laptop with a single monitor. You just have to live with the constraints/limitations of what you are trying to contort yourself into doing on such a system.
I'll let others advise on what they do. But obviously doing a Display Capture and having a visible indicator on the same Display would be an issue. You'd have a similar consideration with an audible indicator of Record/Stream status. So, I'd start with Window capture instead, if possible/practical (depends on use case. personally, I don't use Display Capture)
