The most useful information came from log stream --predicate 'subsystem == "com.apple.TimeMachine" OR processImagePath CONTAINS "backupd"' --info --debug:
2026-03-24 18:12:48.445652-0700 0x3fb9f Error 0x150e6a 533 0 backupd: (TimeMachine) [com.apple.TimeMachine:Mounting] NAConnectToServerSync failed with error: 80 (Authentication error) for url: smb://nsayer@chronos.ca.lan/Backups
2026-03-24 18:12:48.445748-0700 0x3fb9f Error 0x150e6a 533 0 backupd: (TimeMachine) [com.apple.TimeMachine:General] Authentication error (80) - the correct user or password info may not exist in the System.keychain or the server may no longer allow access for this user.
2026-03-24 18:12:48.445805-0700 0x3fb9f Error 0x150e6a 533 0 backupd: (TimeMachine) [com.apple.TimeMachine:Bonjour] Bonjour resolution not possible for 'smb://nsayer@chronos.ca.lan/Backups'
2026-03-24 18:12:48.445833-0700 0x3fb9f Info 0x150e6a 533 0 backupd: (TimeMachine) [com.apple.TimeMachine:Mounting] Attempting to mount alternate URL 'smb://nsayer@chronos.ca.lan/Backups'
2026-03-24 18:12:48.616991-0700 0x3fb9f Error 0x150e6a 533 0 backupd: (TimeMachine) [com.apple.TimeMachine:Mounting] NAConnectToServerSync failed with error: 80 (Authentication error) for alternate url: smb://nsayer@chronos.ca.lan/Backups
2026-03-24 18:12:48.617084-0700 0x3fb9f Error 0x150e6a 533 0 backupd: (TimeMachine) [com.apple.TimeMachine:General] Authentication error (80) - the correct user or password info may not exist in the System.keychain or the server may no longer allow access for this user.
2026-03-24 18:12:48.617425-0700 0x3fb9f Error 0x150e6a 533 0 backupd: (TimeMachine) [com.apple.TimeMachine:General] Authentication error (80) - the correct user or password info may not exist in the System.keychain or the server may no longer allow access for this user.
2026-03-24 18:12:48.617790-0700 0x3fb9f Info 0x150e6a 533 0 backupd: (TimeMachine) [com.apple.TimeMachine:MountedDestinationManager] Error Error Domain=com.apple.backupd.ErrorDomain Code=29 "(null)" UserInfo={NSUnderlyingError=0xb0000077ec54090 {Error Domain=NSOSStatusErrorDomain Code=80 "EAUTH: Authentication error"}} not retryable.
2026-03-24 18:12:48.617937-0700 0x3f7ea Debug 0x150e6a 533 0 bac
Now, I am really, really, REALLY, REALLY, sure that both the password to connect to the NAS and the password to unlock the (encrypted) volume are correct. I have deleted and recreated the TM connection in the TM control panel approximately 39,241 times and it doesn't help. And this same thing happens on 5 different macs connecting to two different Synology NASs all running the latest DSM.