One major disadvantage of this approach is that if you are coming in with a bulky bag, or more (hauling carry-on luggage in an airport for instance), you have sharply limited space within to maneuver you and your stuff to a position where you can lift your skirts and do what you came here to do. Every time I go to a public bathroom with stall doors that open outward I am delighted.

IIRC the Dallas airport had large selves in the wall behind the toilets for luggage. It was extremely convenient.

You are forgetting about pregnants and bigger people. They usually have a hard time getting into the stalls.

Why don't someone design doors that can be both pushed and pulled, and locked

I guess this exists at some places

Plus a mechanism that prevents the doors from opening too fast outwards (so they won't hit anyone)

The bigger reason is that space is usually tight in bathrooms and you don't want to slam the door into someone waiting outside when leaving the stall.