If a backup does not exist in the backup history of that database on that server, as seen in that server's MSDB, then the backup will not be available for restore and will not show up as an option in the 'Backup Content'. There is a text note on the 'Specify Backup Source' screen/page that says "Note: Backup content will be read from database backup history"
It doesn't matter where the historical backups are stored, such as for example, many different backups all stored together on one shared drive. What matters is which LiteSpeed instance / SQL Server was the one that actually created the backup.
The TSQL in the 'Additional Information' section below, if you run that on this SQL Server's MSDB, you will see the same results that LiteSpeed returns. That TSQL is the same query that the LiteSpeed 'Restore Wizard' runs to populate the list of databases available for restore.
We have had a user who would like the Restore Wizard to query the LiteSpeed local database instead of the MSDB. LiteSpeed is querying SQL Server and since that database is new to SQL Server, it doesn't show any backups, even though they are there. That LiteSpeed instance should be looking at history for LiteSpeed to get its history of previous backups.
We have logged this as enhancement request LS-1117: "LiteSpeed query the LiteSpeed local for backup history when using the restore wizard".