So I looked into the issue that half the spots suddenly stopped working, and eventually found that the data line coming into the first spot in the string that no longer worked was constantly high, instead of usually low with the occasional burst of traffic. Looking further into THAT, I found the last LED in that spot looked suspicious:
Replacing that spot solved the issue, for now. But I do worry this has something to do with the overall design, which might mean it’s a matter of time before the next one burns out.