Thanks for your kind reply. I fixed it by reconverting the files with Handbrake. It is now working, so I assume the issue was with my files and not with Picard.
There is a related long-standing issues though about MP4/M4V tags. Let me know if you or someone else could have a look and I am happy to provide sample files or additional context:
Artists/Albumartists tags not supported for MP4/M4V