macOS Catalina requires additional security and privacy permissions to be configured in order to add a local screen as an input. This does not apply to Windows or earlier versions of macOS.
When you first attempt to add a local screen as an input, macOS will prompt you to allow Livestream Studio to record your screen. You should allow this permission or else you will not be able to bring your screen in as an input source and you will instead see the Defunct error notification.
To change this permission, go into macOS' System Preferences and open Security & Privacy.
Navigate to the Screen Recording permission and select the checkbox next to Livestream Studio. You may need to click the lock in the lower left corner to save the changes.
Relaunch Livestream Studio to add your screen as an input.