As a bronze, I get to drop my bag at the Club World check-in (at JFK and MAN usually), and more often than not the agent puts a priority tag on, perhaps because that's what they always do for J pax. I agree that the tag seems to have no effect on when the bag appears.

IME my bag always seems to be one of the first on the carousel or one of the last. My guess is this is because my itinerary almost always involves a connection at LHR, so possibly the connecting bags get put in a container at one end of the set so it always gets loaded/unloaded first or last and never in the middle.