Yes, the force of pressure is equal all the way around. I'm not sure how you are coming to the conclusion though that the force on the rim from the tire pressure is equal to zero. The tire can be abstracted to be a perfect membrane (holds tension but cannot support a bending force), but the rim most definitely cannot. The rim opposes the air pressure through its mechanical structure.

EDIT: I think I see what you are saying. I think if you include the load and the reaction force from the ground in your analysis, you'll find that all thing sum out to zero net force. But instead of just air pressure loads, you have external loads as well. The tire deforms to support the external loads.