I discovered this as part of WT-12478 where we're requiring live restore to create source directories via backups, not copy-paste of the folder.

When we use backups the WiredTiger.backup file will be migrated into the destination and then deleted when the backup completes. This is fine during the migration period as we have a stop file, but when migration completes we'll clean up this stop file. If the then restart WiredTiger it'll open in the COMPLETE phase, but our fs_exist logic will find the WiredTiger.backup file in the source folder (as we don't have a stop file) and try to start a new backup even though the backup has completed.

The issue that needs fixing is when we're in the COMPLETE phase fs_exist should only consider whether a file exists in the destination. This ticket makes that fix and also adds unit testing for fs_exist