There are a few things you can try to fix the issue with the Touch ID option disappearing from your logon and settings.
1. Restart your iMac. This may be a simple fix that will resolve the issue.
2. Check for updates. Make sure your iMac is up to date with the latest software updates. Apple often releases updates that fix bugs and improve performance.
3. Reset Touch ID. If restarting your iMac and checking for updates didn't fix the issue, you can try resetting Touch ID. To do this, follow these steps:
1. Go to System Preferences > Security & Privacy > Touch ID.
2. Click the Remove All Fingerprints button.
3. Follow the on-screen instructions to add your fingerprints again.
4. Disable Touch ID. If you're still having trouble, you can try disabling Touch ID. To do this, follow these steps:
1. Go to System Preferences > Security & Privacy > Touch ID.
2. Uncheck the box next to Use Touch ID for Login.
If you've tried all of these steps and the issue is still not resolved, you can contact Apple Support for help.