Isn't this less about being 'horizontal' or 'vertical', and more about the fact that we display information on flat areas, and about different disciplines relating to segregation of 'in-progress', 'pending', and 'archived' material?
If I have five pieces of paper relating to a task that needs to be done, but which I'm not working on right at this moment, I'll stack them up simply because that leaves more space for the task I am working on right at this moment. Now, that stack may be 'horizontal' on the desk, or it may be 'vertical' in a filing cabinet. But it's simply a space-saving measure.
His main problem seems to be that when he files pending stuff away, he forgets to look at it again. But this isn't about vertical versus horizontal thinking -- it's just about bad document management. If you just have a monolithic filing cabinet that contains both ancient archived documents, and material relating to pending tasks, then yes, you're going to get confused and forget about things. If, on the other hand, you have one file as an archive, and one file that you know is for pending material (or an in-tray; or just a special pile on your desk), then you'll remember to check it.