I know the correct option is to hang a single drywall board horizontally over the doorway, and then cut out the door. But just how terrible would it be to hang horizontal boards on either side, and then a single thin piece over the top? The sides would be small butt joints. Would it really be the end of the world as some forums are making it sound?
Every building settles. A common place where drywall cracks due to settlement is at the top of doors. Minimizing joints at top of doors is highly recommended.
A razor knife is a great tool for cutting sheetrock. I would guess any major settling has already happened. Reducing joints over doors will reduce cracking of a joint but with severe settling the sheetrock can crack or pull through the screws I don't think you are risk with your home.
Small but joints above each corner of the door ( properly taped )are not going to be to difficult to feather out and make less noticeable. You could choose to put one but joint somewhere over the door so you only have one to deal with and it will not affect the installation of trim at the corners.
More importantly It appears your door is not hung properly. In the photo it looks like the jamb (door frame) is set flush with the wall framing so that you do not have anything to butt the Sheetrock to. The jamb should sticking out past the framing (stud) by the thickness of the Sheetrock so that when you put trim around the door it sits on the Sheetrock and overlaps onto the jamb.