ITunes can't locate some files that are right where they should be

Question

I'm using iTunes 12 for Windows.  I keep a huge iTunes library at home on a mapped NAS share and the same library at work on a local drive.  Every week on Monday I sync my entire iTunes folder from home to an external drive and then bring that external drive to work and sync it to my local drive there.  (Note: This is a one way sync; I never sync in the other direction.)  This system has worked flawlessly for six years.
Within the last six weeks, I've started to get error messages on my work system saying that many (>1600) media files of various types cannot be located.  I assumed corruption was the issue so I deleted my entire 500GB iTunes folder and re-copied everything back over.  Not just the media files.  Everything.  The problem persists and I cannot figure out why.
I have confirmed that iTunes on the work computer is referencing the correct iTunes library file.  I have confirmed that iTunes is pointing to the correct iTunes folder.  I have confirmed that the "missing" iTunes files are right where they should be.  (I have always had iTunes set to move imported media into the iTunes folder so there is no stray media floating around on either system.  And I have iTunes set to organize the media.)
It is worth noting that I started seeing this around the time when I upgraded to iTunes 12.  When I upgraded to iTunes 12 I had to change the path to my home library from "\\<NAS>\iTunes\iTunes Music" to "M:\iTunes\iTunes Music" due to a bug in iTunes that created iPhone syncing issues if the iTunes library is on a network drive.  But I can't figure out why this would matter.  The path on the work computer is correct and all media files are there.
Ideas? I just don't understand how deleting everything and starting over doesn't solve the issue since there is NO issue at home.

Answer 1

Thanks again for your help with this, tt2.  (I keep wanting to call you "Alan".)
I totally botched the repair process, including my library and media collection, four different times.  If not for the fact that I had a complete, up-to-date copy of my library and all media files I would be completely scr3wed right now.  I'm going to outline the mistakes I made for the benefit of those who might reference this thread in the future.
A big part of my struggle was the underlying assumptions I was making about how the iTunes database is maintained and tracked.  It is not the same as the digital asset managers I use for work—Lightroom, for example.  I really had to quit thinking for myself and pay attention to what tt2 wrote about the way the iTunes db works in the link he provides earlier in this thread about making split libraries portable.
The biggest mistake I made is failing to understand that after you move the iTunes library files (but not the media files) to a new iTunes drive or directory outside of the existing iTunes folder and then shift-click to open that relocated library in iTunes you have to set the iTunes Media folder location (in preferences>advanced) to the empty iTunes Media folder in the NEW iTunes folder.  You do NOT set it (or leave it) pointed to the old iTunes Music/Media folder where your files [may] actually reside before consolidation.  This mistake can make a bad situation much worse. 
Finally, I did not have a split library in the sense that my media files were outside of my iTunes Media folder.  Because of this, when I incorrectly pointed the new iTunes library to the old iTunes Media folder and then consolidated I created 9,936 duplicate files.  I had a complete back up which made eliminating the duplicate files a simple process.  Without the backup I would have been in trouble.
So...to fix my problem I followed tt2's instructions and did this:
Backed up my entire M:\iTunes folder including libraries and media files.  If everything goes well, this backup will not be touched.
Created a new A:\iTunes folder plus an empty A:\iTunes\iTunes Media folder.
Moved everything from M:\iTunes to A:\iTunes EXCEPT for the iTunes Music* folder.  (*It was the new iTunes 9 and later hierarchy but with the old name.)
Shift-Clicked to open iTunes and selected the iTunes library inside A:\iTunes.
Once iTunes was open I went to Edit>Preferences>Advanced and made the empty A:\iTunes\iTunes Media the location of my media.
Went to File>Library>Organize Library and choose "consolidate". After hours of file copying, my issue was resolved. A:\iTunes is portable.
Confirmed that the new A:\iTunes library correctly links to all media in the A:\iTunes\iTunes Media folder and then deleted the M:\iTunes directory with the old M:\iTunes\iTunes Music folder inside.
NOTE: It bears repeating that the problem solved was that I could not move my old M:\iTunes folder to a different workstation w/o breaking some of the links.  However all the links worked fine in the M:\iTunes location.  If this were not the case then this solution will not resolve anything.