After reading
Google's explanation of how licensing works and doing some further tests, I'm concluding that it's not useless.
If licensing is enabled, then the game tries to connect to Google's servers and query whether
the user of the device ever purchased (or, if free, simply downloaded) the game from Google Play. If there's no internet, then the game prompts you to connect every few seconds, making it unplayable. If the device's active user account (i.e., the Google account set as the device owner) has never downloaded the game from Google Play, it does the same thing. If, however, the device owner ever purchased the game from Google Play, then the license is considered valid henceforth and there's no nagging.
I was initially a bit thrown off by the fact that (1) my APK was sprayed across a bunch of sketchy internet sites, and (2) I was able to download a couple APKs and play my game using them. I now understand that (1) at least in many cases, the APK wasn't altered, and (2) I was able to play my game because my phone's user account had previously downloaded the game from Google Play. When I created and used a brand new Google account, I was appropriately nagged.
Of course, I'm not downloading every one of the APKs that appear to be on the internet, but I'm satisfied for now that the licensing works. I believe that anyone who downloads it from those sites won't be able to play the game. Whether that's good or bad depends on your opinion on how to respond to piracy.