As MC changes tracks, it will increment the play count and will change the Last Played Date. Those values get written to MC's database. If you also have the "update file tags" switch turned on, then MC will try to write those values into the file metadata tags also.
Normally I would say that your files are read only or something else is preventing writes to them. Something like security software or something. But you have said that they are not read only.
You also indicate that you can change the tags manually without issue. Can you make a change to a file and then check that file with an external tag reader like KID3, MP3TAG, VLC, etc? It would be instructive to see if the tags really are changing inside the file or not.
You might also go examine the file itself on the very next failure you get. Does the Last Played Date reflect the play that just happened? Is that particular file actually writable? it's possible that your drive and your top level folder are read/write, but individual files are read only.
You could "shotgun it" and run a recursive permissions change on the entire drive. I wouldn't do that until I was sure, but it's one approach.
Best of luck,
Brian.