Do you have another monitor that you can plug the HDMI cable into? This would tell you if the iMac is capable of detecting a display through the HDMI cable. Helps split down where the problem is.
Also, it looks like you're running Ventura. That's added a few extra twists. External devices connected need to be 'approved' the first time they are used. 13.1 had some issues where it just ignored devices. Make sure you update to the latest OS, 13.2.1. You can then check in the Privacy and security settings under Security to always allow accessories to connect.
They also have hidden the 'detect displays' option. In the displays setting window that you've shown, hold down the option key and the detect displays button should appear. Click that and see if it finds it then.
If none of that works, downgrading to Montery might be a solution as well.