I hope this guide can help you for OBS not showing up in the System Preference -> Security & Privacy -> Privacy -> Screen Recording.
The problem might be caused by "System Integrity Protection" being disabled in your computer. In order to enable it, you can check out this guide https://totalfinder.binaryage.com/enable-sip. I've checked a lot of threads of people asking for why they can't see Screen Recording Permission not showing up in the tab, and I hope this method can help you.
REMINDER: Enabling System Integrity Protection will RESET your computer's Privacy section, so ALL of your apps will ask for permission for "Accessibility", "Microphone", etc. You'll need to enable them again IF NEEDED
The problem might be caused by "System Integrity Protection" being disabled in your computer. In order to enable it, you can check out this guide https://totalfinder.binaryage.com/enable-sip. I've checked a lot of threads of people asking for why they can't see Screen Recording Permission not showing up in the tab, and I hope this method can help you.
REMINDER: Enabling System Integrity Protection will RESET your computer's Privacy section, so ALL of your apps will ask for permission for "Accessibility", "Microphone", etc. You'll need to enable them again IF NEEDED