"File (Field) Paths" and "Media" modules: files not saved in the intended folder
No definitive answer so far, but I should note that I have been able to make this option work again simply by enabling the (very intimidating) option "Active updating" in the field's settings. (So in admin/structure/types/manage/[content-type]/fields/[field-name])

The "File (Field) Paths" developer warns that "This feature should only be used on developmental servers or with extreme caution", but since it's the only solution I've found so far, I'm going to keep this option enabled for now.

Hope this helps someone.