To answer my own question, no, the iOS/iPadOS updates did NOT mess up my devices ability to see my AirPrint server.
The day after I posted this thread, my Debian machine suddenly glitched badly. It basically forgot the printing system (CUPS) that I’d just reinstalled. Couldn’t find anything, couldn’t even run the web interface. Aha.
So yesterday and today I reinstalled the whole Debian OS. It turns out that somehow there was a mix of x86 and x64 packages installed on the thing, and they most likely “did not play nicely with each other.” Oh, and there have been a few OS updates in the past few weeks, so I’m thinking that the updates just pushed the system over the edge and it started to fail.
This morning, I got everything (that I wanted) reinstalled, and tested. And before I dug in to figure out if I needed to spend another day configuring stuff so AirPrint would work, I gave it a shot. And sort of like the documentation says, “it just worked.”
I’m going to try to make a clean clone of this setup, so if something else goes “snap,” I can just reinstall from the clone.