Restore job run successfully without error but actually not applied during log ship from older version to newer one
You cannot restore using STANDBY from a lower version. If that is what you mean nu "open these standby databases after a long time).

STANDBY implies a recovery, which would need to physically upgrade the database. An database cannot be downgraded again, which in turn would be necessary when restoring a subsequent log backup.

Timestamp of database files are irrelevant. Don't bother with those. They aren't updated when information inside the file is modified.

You can keep the database in a restoring state for a long time, i.e., log shipping doing restore using NORECOVERY. But you need to have a schedule between the three agent jobs so that the restore job actually have the backup file available. So, for instance backup every hour, copy file every hour, restore (NORECOVERY, of course) every hour. And then delete the backup files after, say 1 day. That way the files neeeded for the restore will be on disk.