Incorrect battery makes it seem this is broken
Summary answer: CR2016 physically fit, but won't work - double-check that you have the right battery!
Hopefully not too naive a question, and I have the answer: using a CR2016 instead of a CR2032 will make it physically feel as if you've added a fresh new battery, but it won't work.
I ended up coming on here to try (and then succeed) to open the device up, only to try a battery swap to prepare to test anything I might do to fix it, and then find that everything was actually fine!
crwdns2934109:0crwdne2934109:0