Copying by drag-and-drop in the Finder should work for any file, regardless of the capabilities of Monterey QuickTime Player.
Verify the MOV file size. Verify the free disk space on the thumb drive. The file size should be non-zero. The free disk space on the target disk should be larger than the file size. What is the MOV file size?
Consider using VLC or IINA for media files that this QuickTime version has problems with. VLC is also a good advise for your contact, as it is available for many platforms.
If you want, you may be able to convert the file to e.g. MP4 with H.264 video and AAC audio, so that QuickTime Player likes it better. But when using VLC, that wouldn’t be needed.
… created on my iPhone about 5 years ago.
More importantly, which model iPhone was that, with what version iOS, at that time? That determines the format of the media file, much more than the recording date.