Priority only affects Resource allocation?
 

The manual says: "If two tasks are assigned to one resource at the same time, the task with higher priority gets to use the resource first."

Is that the only effect Priority has? I was hoping that via Leveling, it would push tasks with higher priority closer to the present date.

Yes, that's what it means. When you level, if two tasks require the same resource (and don't have other constraints affecting their schedule), the task with the higher priority will be scheduled earlier (closer to the present date).

(If the tasks don't share a resource, then they'll both be scheduled as early as possible, and there's no reason to consider their priorities.)