I suppose you've never had a misconfiguration corrupt all your data before? Yeh, it's on the disk, and it's just a program reading/writing to that disk... but even if those programs are cattle, those disks sure as hell are not.
Which is why pretty much every Postgres k8s operators includes a backup solution that is fully integrated.
I much prefer to be able to treat the database instances like cattle, while the conceptual "cluster" and its associated backups should be treated as a pet, as you said.