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The movement that you are experiencing is normal as the cylinder is surrounding by a spring that provides a small amount of in-and-out free play. The lock cylinder shown in your ebay reference shows that spring. The spring is supposed to assist in clearing the catches when the key is turned. The one on ebay in #4 above lacks such a spring.
Yes, the wheel is secured to the spare wheel bracket with three lug nuts and the purpose of the cover is to prevent unwanted access to those lug nuts.
Scroll down on your ebay listing to the 17th (or so) item, which is a complete deluxe '36 spare tire cover and lock. If you go to the photos of the back side you will see the back end of the lock assembly. Note the little catches on either side of the bottom of the lock cylinder. They are what engage the spare tire bracket on car's back panel and by turning the key of the lock they are supposed to retract so that the cover face plate assembly can be removed.
Perhaps the catches are hanging up as it may be that your 12-to-2 rotation of the key is not enough. Sorry, I am away from my '36 so I cannot verify how much the key has to rotate to retract the lock catches completely.