マイケルさん、
That's an interesting observation!!
I think how you "position" the "main purpose" is very interesting, but it also depends on person. I mean, the door could be there so that people can open the path through, OR, it's there to close the space. However, books and umbrella's case, it totally make sense!! Well done!!
One thing I didn't mention is "the feature of itself". For using "tojiru", it's not only about closing, but also the action of closing has to be with the feature of the object itself. This is why you can't use "shimeru" for eyes, but it has to be "tojiru". On the other hand, when you use "shimeru", the object itself usually cannot take that action (someone would do that).
Hope this helps too!
Natsuko(奈津子),
Team JapanesePod101.com