There's a fundamental problem with the way public restrooms are used. A lot of restrooms still have doors that require opening by hand to get inside one. When someone needs to wash their hands because their hands are dirty, they're going to have to touch the doorknob with those dirty hands because the place to wash their dirty hands is on the inside.

Secondly, when most people need to go into a stall for a #2, they don't wash their hands before they use their hands to open the stall and then do their thing inside. They wash their hands after. Another aspect is the use of toilet paper from one toilet paper roll. You grab it to get some paper to clean your ass, then you touch it again some more. Then you go up, pull your pants up and then open the stall with your hands, then you finally wash your hands. Bathroom stalls can often be cramped too so some people can't operate without possibly rubbing against the stall walls, which can be pretty gross. Another flaw is that toilets get flushed with the toilet cover open. Imagine all the shit particles that get shot all around the stall.