You can get those [not being able to update] messages even though there are no broken links... searched a workbook many times, but there's none [not even in named ranges that contain a dynamic formula].
Of course, linking to files with the same name is asking for trouble, and the last warning screenshot confirms so. Would you ever open that archived copy, it will substitute links to point to itself.
One pit-fall is using certain methods of index/match with offsets to fetch data from an external book. You could end up with #N/A evaluations, subsequently followed with warning of errors.

EDIT: An example what does work, where the name\location of the file can be changed by making indirect look at a secondary field to get the file name [in text form]. Sample

=INDIRECT("'[WhateverFileName.xlsm]SheetName'!RangeCellName")

The last error message was defintely a mistake. I wanted to leave it out as it's simply fixable. I m working on two cases simultaniously so sometimes i forget to close files in one of the two cases.
The remaining error messages are still hard to get rid of though.

As i said initially, the problem about the File>Info>Edit Links to Files method is that i have to do it manually for every single link. And Word won't allow me to just mark everything and change it. It only alllows a limited number at a time.
That's why i was looking to see if there was a smarter way of handling it.