It seems I have been under the impression this whole time that the OTA updates would happen automatically. I was wrong. I searched around and found this guide to enable automatic OTA updates in ZHA and am currently updating one of my switches.
My switches were on firmware version 0x01020200, which I think corresponds to v2.00, which is older than November of last year, so I was pretty far behind in updates. I’ll update this thread after the update is complete and I get a chance to test.
P.S. I also just discovered this thread, which I missed in my prior searching, which seems to describe my issue.