I figured out the problem. I can't fathom why, but using an SSD for the internal drive was causing the problem. I had two spare 500GB 2.5" SATA3 SSD lying around and thought it'd immensely speed up copying games over plus much faster seeking times.
After the above issue happened, I tried the second SSD and had the same issues. It was preventing certain file copies too. I eventually went to an old 250GB 2.5" HDD I had lying around as a last resort and everything works fine now. Bear in mind both the SSDs were known good and working, recently pulled from my PC due to upgrading my M.2 drive.
Just thought I'd reply to my own thread with the solution for anyone in future that might dig this up. Absolutely insane that a far superior and faster drive causes problems! My gut feeling is that the console was made when SSDs were a pipe dream and they never intended to release a 360 with one, so programmed the console to spit out an error when file transfers complete too quickly or something. Maybe there's a workaround or a fix out there somewhere but I've loaded up the HDD now so will probably stick with it.