Continuity Camera on a Mac mini
It looks like the Mac needs a "separate" camera before you can select the iPhone's camera. If you don't have a camera connected to the Mac, you will not even see the iPhone camera as an option.
I have previously upgraded a MBP 13" to Ventura and the continuity camera worked flawlessly there. When I upgraded my Mac mini 2018 to Ventura, I could not see the option to use my iPhone's camera. It was until I connected a separate webcam which I use on my Windows computer to the Mac mini, that's when I was able to switch to the iPhone camera.
Probably need to add that tidbit in the Additional Requirements for the Mac minis.
Mac mini (2018)