UPDATE: FIXED
I figured this out, though my use case is probably more specific than most.
I was using my iPad with a Kuxiu iPad Stand: a magnetic stand that support the Smart Connector to charge the iPad while docked.
I also have an LG ultrawide display.
My MacBook was on the left side of the display. The iPad was mounted on the right.
Whenever I took my iPad off the magnetic stand, Universal Control suddenly started working.
When I docked it again, it would work briefly, then drop.
I decided to test this and move the iPad stand to the left side of the desk above the MacBook.
Universal Control is now working flawlessly.
It seems the magnetic stand is thick enough to degrade the signal to the Mac *just enough* to break the connection between it and the iPad.
I suggest removing your iPad from any docks/Magic Keyboards/cases and holding it near the Mac to see if the problem persists.
Sorry this probably won't be more helpful to others.