I was suspecting a broken file system on the Time Capsule, but needed proof. After yet another mysteriously cancelling verification step, I found https://superuser.com/q/1126990/7401 which ended up in
log show --last 3d --style syslog --predicate 'senderImagePath contains[cd] "TimeMachine"' --info|less
which in turn let me find this log line:
2021-04-05 00:48:38.359222+0200 localhost backupd: (TimeMachine) [com.apple.TimeMachine:General] fsck returned an unknown termination status: 8
So, the file system check program reported an error message but Time Machine didn't let me know. Apparently 8 indicates unrepairable error, which is a rather bad sign. I then mounted the "Data" drive holding the backupbundle file on Time Capsule manually, and as Disk Utility did not want to mount the backupbundle, I found https://apple.stackexchange.com/a/39842/3157 resulting in
sudo hdiutil attach -nomount -readwrite /Volumes/Data/XXX.backupbundle
Now the Time Machine backup shows in Disk Utility, and I could run First Aid on the drive which slowly but successfully repaired the filesystem.