Why do men wash their hands only after they pee?

I have always wondered why we do certain things a certain way even though it doesn't make sense.  One such example is the practice of men washing their hands only after they pee in a public washroom.  Why only after?   Why not before?  Here's my train of thought ...

You wake up in the morning and presumably, you go take a shower and clean yourself.  Your entire body is now clean, especially your genitals as they are covered and protected by your clothes.  Your hands meanwhile, are touching door handles, food, money, etc. all of which add many more harmful germs and bacteria to your hands, than your genitals.   In fact, your genitals are much cleaner than your hands at this time.  So why go into the bathroom and contaminate your genitals with your hands, and THEN wash your hands after?  If anything ... wash your hands BEFORE touching your privates, and yes certainly after as well, so as to respect the next person whose hands you will shake.

But ... if you have to pick one of Before or After peeing ... pick Before and keep your privates safe!