Universal Control is not working in recent OS updates

Users report Universal Control feature to be broken by macOS 13.3 and iPadOS 16.4. Recent updates seem not to fix the issue either.

Multiple issues related to Continuity seem to be affected. But most reports are in regards to Universal Control.

Users also variously report problems with Handoff, Universal Clipboard, and Auto Unlock and Approve with Apple Watch. Based on online reports, the main Continuity feature to be negatively impacted by the updates appears to be Universal Control.