It's worth mentioning that 閉じる has a spatial nuance to it. When I studied Japanese as an undergraduate, we were taught to differentiate 閉める and 閉じる by this nuance. For example:
- プログラム、目、扉 ー ひらく、とじる
- 瓶、パッケージ、ドア ー あける、しめる
What we were taught is that if it involves opening something outward/to the side (and vice versa if it were being closed), like a book or your eyelids, you use 開く・閉じる.
Now this is not a hard and fast rule, but it is enough to illustrate the idea that when you close a door with 閉じる, there is a nuance of pulling it shut. The image for me is a double door or a sliding screen, which, when standing in the doorway, must necessarily be pulled closed, and conveniently, in the case of the sliding screen, pulled open.
This might explain why there is no entry for 押し閉める, and why my IME doesn't recognize it. There's no essential need for it.