I have rented from the aisle for the past three years and I have never received a contract in hand. The only time I ever receive paperwork is when I rent from the desk at a local branch.

Your concern over being pulled over by the police is a bit of an extreme situation... well, maybe not if it happens to you often. However, if there is ever a case where you are stopped by the police, they will just ask you to call National and have someone confirm your rental. I had to do this once when I had my car towed and all I was required to do was show them my executive elite membership card and to call national.