Nearscreen Receiver
The desktop app your iPhone streams to. Free, for Windows and macOS.
Open source, on purpose.
Nearscreen’s whole promise is that your screen never leaves your local network. You don’t have to take our word for it — this receiver is all the code that ever touches your video, and you’re looking at it: github.com/nearscreen/receiver.
Capture it in OBS
The window keeps a stable title — a regular Window Capture just works.