Your second paragraph is absolutely correct, yet it leads to a circular argument, because it does not tell you anything about how correct your paper is. In fact

Your second paragraph is absolutely correct, yet it leads to a circular argument, because it does not tell you anything about how correct your paper is. In fact, in that experiment, the probability of your paper being among the correct ones is the same quantity as the actual population density. You cannot derive any meaningful information from assuming you are above,about,or below average.