Well, here's some good news. I've updated quite a few pre-releases of CDVR server since I started this thread, and also a few point releases of macOS Sequoia on this Mac mini, and I have NOT encountered these two Privacy & Security pop-ups again. There were pretty frequent a few months back, very noticeable, and at least twice, it proved to be a literal showstopper. And now, I'm not seeing them at all, despite doing manual updates regularly and sometimes remotely, just as before.
I even upgraded my Mac mini to macOS 26 Tahoe, then to 26.0.1, and the Channels DVR server is still launching automatically without any additional or unexpected pop-ups after restarts or reboots. So I think that was just an unfortunate bug with some earlier versions of Sequoia, an aberration perhaps? Fingers crossed that it's a thing of the past!
Felt like the tables had turned for awhile: