Sorry, I was wrong about the courts, but at first it is not even handled by a court or prosecutor at all, only by the administration. If the potential fine is not accepted it then indeed goes to criminal court.

Well, yes, Ordnungswidrkeiten are "Straftaten light", if you will, so consequences are generally less severe, the process has a shortcut for undisputed cases, ... but the essence is still that the state makes a member of society experience some consequences for not following the law, and thus in particular the same constitutional protections apply.