iSight Camera not detected on Mac mini (Ventura)

1) Something is wrong with the model and/or macOS info that you have provided.

see > Identify your Mac mini model - Apple Support

the 6,1 2012 Mac mini can run up to macOS 10.15 Catalina.

the 7,1 2014 Mac mini can run up to macOS 12 Monterey.

the 8,1 2018 Mac mini can run up to macOS 14 Sequoia.

2) The 6,1 2012 Mac mini is the only one of those that has a FireWire 800 port.

For it you need to be running 10.15 Catalina or earlier

A FireWire 600 to FireWire 800 adapter.

3) If you have the later 7,1 2014 Mac mini, it has a TB 2 port instead of a FireWire port.

For it you can be running up to macOS 12 Monterey, but not 13 Ventura.

You need an Apple Thunderbolt 2 to FireWire 800 adapter.

Plus a FireWire 600 to FireWire 800 adapter.

I just did a quick little bit of research and it seems that some of the support for FireWire was dropped with the release of Ventura, specifically FireWire Core Audio. This affects devices such as your iSight Camera, but would preserve access to storage devices which are arguably more important to more users.

If you wish to continue using the iSight cam, then you might consider reinstalling the previous OS, Monterey, on an external boot drive to evaluate whether it is fully supported in that OS. If so, then you could reinstall Monterey on your mini and live with that.

Otherwise, you might save yourself from the hassle and simply find a modern USB replacement for the old webcam.

As a potential alternative: Continuity Camera: Use iPhone as a webcam for Mac - Apple Support

maybe it's too old to be accepted

I had usb webcams only work in photobooth because the facetime app view it as too old and poor quality.

it was discontinued in 2008 which is somewhat many years ago.