It's fairly obvious why stopping a service while backing it up makes sense. Imagine backing up Immich while it's running. You start the backup, db is backed up, now image assets are being copied. That could take an hour. While the assets are being backed up, a new image is uploaded. The live database knows about it but the one you've backed up doesn't. Then your backup process reaches the new image asset and it copies it. If you restore this backup, Immich will contain an asset that isn't known by the database. In order to avoid scenarios like this, you'd stop Immich while the backup is running.
Now consider a system that can do instant snapshots like ZFS or LVM. Immich is running, you stop it, take a snapshot, then restart it. Then you backup Immich from the snapshot while Immich is running. This should reduce the downtime needed to the time it takes to do the snapshot. The state of Immich data in the snapshot should be equivalent to backing up a stopped Immich instance.
Now consider a case like above without stopping Immich while taking the snapshot. In theory the data you're backing up should represent the complete state of Immich at a point in time eliminating the possibility of divergent data between databases and assets. It would however represent the state of a live Immich instance. E.g. lock files, etc. Wouldn't restoring from such a backup be equivalent to kill -9 or pulling the cable and restarting the service? If a service can recover from a cable pull, is it reasonable to consider it should recover from restoring from a snapshot taken while live? If so, is there much point to stopping services during snapshots?
Not a VM. Consider the service just a program running on the host OS where either the whole OS or just the service data are sitting on ZFS or LVM.
This is one of the reasons Docker exists.
And I'm using Docker, but Docker isn't helping with the stopping/running during backup conundrum.
It should work that way. If you use the recommended Docker Compose scripts for immich, you'll notice that only a few volumes are mounted to store your data. These volumes don't include information about running instances. If you take snapshots of these volumes, back them up, remove the containers and volumes, then restore the data and rerun the Compose scripts, you should be right where you left off, without any remnants from previous processes. That's a pro of container process isolation