Unfortunately Time Machine when it migrates between drives loses its ability to be accessed via the standard Time Machine interface. That isn't to say that you can't take the individual archived folders and navigate them after being transferred.
A couple alternatives....
Install a newer or the same Mac OS on the empty LaCie drive, and during the installation on the LaCie drive, use the setup assistant to import the Time Machine backup from the other drive.
Use Shirt-Pocket Superduper to copy just the backupdb folders from one drive to another. It will maintain the permissions and structures of the backup.
A last ditch alternative loses the permissions of the files, and just gives you flat files with 100% read/write permissions. Finder's File menu -> Get Info has an Ownership & Permissions section of the Get Info of the selected item. When it is a folder, you can change the permissions of all the enclosed items. Recovery of specific libraries of photos and playlists from one drive to another may not proceed that smoothly. Although both Music and Photos allow importing without managing the library through their preferences once the files are copied to a new drive. This is not a great solution though if you were trying to migrate licenses from an older drive to a newer drive before the old drive failed as the licensing folders and files typically must be in specific administrative folders.
There used to be a software by Arbysoft called Batchmod that did this, and could accidentally change the entire system to the wrong permissions. Arbysoft no longer exists. If you are careful and can find the software without any advertisement or optimization tools, it can be very handy.
Devon Technologies EasyFind allows searching for files by name and date without having to worry about Spotlight indexing.
https://www.devontechnologies.com/apps/freeware