Premiere Pro 14.6 shows clips w/ "danger stripes" after moving media to another drive

Just for clarification, are you opening the project with the source footage still in the original location?  Or are you opening the project and the files on the backup (maybe even on a different computer)?

This will take some patience, but it's worked for me in the past.  Open the project file with the media offline and do a File > Save As.  Append something to the file name like "_please_work".  When I did this, I moved the project file from the external drive to the desktop and then unmounted the external drive.  Then I opened the project making all the media offline and then put the Save As also on the Desktop.  I then reconnected the external drive, opened the Save As project file, and took the time to relink the media, checking it as I go (so, not all in one pass).  

For some clips, I would use Replace Footage instead of Link Media.

Since you mention that the original sequences also have trouble, it may be worth opening a recent auto-save to spot check it to see if the issue is presenting itself there as well.

Lastly, how did you do the backup?  Manual drag and drop?  A copy utility like Carbon Copy Clone or EaseUS?  And, is the LaCie drive formatted the same way as the original drive?  So, ExFAT going to ExFAT or HFS+ going to HFS+?  Reviewing this may prevent issues in the future.  When I ran into stripes in the clip boundaries, it was a project that had been uploaded to Google Drive and then downloaded and it was mostly AVCHD footage that was not reconnecting correctly.

Please tell us your system specs: OS version, Premiere version, amount of RAM, Hardware specs including graphics card.  was the new drive formatted the same as the source drive and is the folder structure the same?  If you're on the mac, I'd suggest you use a cloning program to exactly match the same folder structure so that if you rename the new drive to match the old drive, everything should relink without issues. 

Usually I see this happen when there is multiple instances of very similarly named media on a drive. Do you have other similarly named clips on your drive? When it goes looking for the clips it can load in the incorrect ones, at least in the situations I've been in like this. I've had to be sure that the files are uniquely named and then relink them.