So I'm apparently one of the few where this was actually an issue with the switch and not a malfunctioning actuator.
Again, my symptoms were:
Pushing in the switch above the license tag does nothing. However, pushing the rear hatch switch on my keyfob worked fine, every time. Symptoms started in early May, so I ordered the actuator kit. After that early May incident, however, everything started working great again until around a week ago, when the switch stopped working altogether.
What I did:
Replaced the actuator tonight, and the problem was still there with no change at all.
I then pried out the switch from the liftgate
more or less like this, removed the external rubber part, and the internal paddle portion just fell out with it. (The paddle in the video was aluminum, but mine was plastic.)
I pushed on the button a few times and nothing worked. I then sprayed the bejesus out of it with some DeOxit spray (from back when I was restoring vintage stereo equipment as a hobby) and the button started working again. Put the paddle back in correctly, put the rubber back on, and it all works fine now.
I'll keep an eye on it, but think I'll probably wind up replacing the switch as the paddle just doesn't seem like it's going to stay in place. Thing is, I still can't really figure out the part number. All the standard online Land Rover dealership parts suppliers seem like they want you to buy the whole entire liftgate assembly. Nearest I can tell, it's either LR015457, which only turns up knockoffs, or LR071910, which turns up no supplier stateside.