@OM2 -- thank you for your detailed response. At this point I would have sacrificed a small goat if someone had suggested it might fix my problem. I followed your steps to the letter, but alas they did not work for me. I even tried twice, the second time restarting the NAS and clearing all connections from there, but the NAS never appears under Network or in the sidebar. I've checked and rechecked the settings in the NAS to ensure Bonjour is turned on and the NAS is discoverable from the network (keep in mind that I never changed any of the NAS settings from before, when my MacBook was discovering it automatically.) Zilch. Nada. No dice.
The only way I can access the NAS now is from Go >> Connect to Server, and that connection has all kinds of problems. It disconnects (as the connection used to from the sidebar), and even when it is working, backups take forever, despite no issue with the network or the NAS. (I can upload/download large files from Synology Drive without similar delays.)
Whatever this bug is, it's deep enough to destabilize all the apps on my Mac when it's on the same network as my NAS (they run just fine when I'm off the network), and it's persistent (nothing seems to work to clear it out and get back even to the original state.) Worse, because of the improper disconnections between the Mac and the shared folders on the NAS, data is being corrupted and lost. As I mentioned in my last post, there are only so many times that TimeMachine can lose connection, mid-backup, or Photos can lose connection, mid-library-update, before the underlying data is damaged and/or lost.