I believe it’s to do with hand cleanliness. When you enter, you push the door as your hands are dirty (maybe shove it with your arm or something), then when you leave your hands are (supposed to be) clean so you pull the door as it’s a nice clean handle to grab.

I don’t get it either, but that’s what I’ve heard as the reason.

So then we need saloon doors that swing both ways?

I actually think this or just a double hinged *kitchen style" door is the best answer