For a small business to implement a process of sending in bills they would have to rework their pricing to reflect the extra hours required, as well as the extra costs inherent in such a system. Bottom line: billing would make their cost go up.
During a session with a student or client you want the attention to be completely on the matter at hand. If the payment is handled online, or a check is presented before beginning the session it allows both parties to be more present in the process. And time is not wasted in you handling the money, or thinking about the money during time you could be working. Bottom line: Paying before the fact makes for a more focused lesson.