I have never seen an elevator not close the doors when the button was pressed. This is really weird. Doors open and close controls work. I think they work 'slowly', but they do work. It may depend on when the system (elevator control) was installed... but, hm. Have to keep an eye on this now! Ridden in a lot of elevators, especially worked in several high-rises, and never had this experience.
Door open buttons do something, sure. But the doors close on their own, after a period of time - and in my experience, that's the same time it takes for the door close button to "work slowly" in every case where I've tried it both ways (which admittedly isn't that many. I'm curious, but not curious enough to spend extra time in elevators with a stopwatch.)