I can't remember the last airport I was at where there was an actual door you had to open. Although that is probably more for convenience w/bags than cleanliness.
At least around here, crowded places started removing door handles after the SAS epidemics on Asia. I am pretty confident it was because of health issues.
I am assuming the airport had stalls with doors, and those doors had latches. I admit I don't travel very widely, but I'm yet to see hands-free stalls.
It shouldn't be hard to order the sequence of steps such that washing your hands is your last activity before leaving the bathroom.
That's true. I've seen these buttons at the Seattle airport, and they were positioned at the door. But that doesn't mean they would have to be there.
Pro tip, always use paper towel to open door when leaving restroom.