I've only seen doorless toilets in large well ventilated spaces such as some train stations. I don't think that would go well inside a shopping mall or a restaurant
Eh, it's unclear honestly. After reading some comments I realised op maybe meant the direction in which doors open, not the fact that you have to open doors. I don't know
Some of the comments might be about stalls, but I think most of them might be about the door to the bathroom itself. Since a person would wash their hand after leaving the stall and before touching that door, and with a pull door you have to touch that handle.
I realized it with this comment since a doorless stall sounds like a nightmare