That's not the question, really. Well, perhaps it is. The answer is "the correct set of rules." In their argument, they make a false assumption about the probability of which urn is more likely...

...and then they use that false assumption. And that's the problem. You are reasoning with a false assumption, so you can't logically say anything about the correctness of your conclusion.

The truth in this case is that, reasoning in the same way that I do with urns (which I hope is correct), there is an equal chance of the urn containing any number of balls greater or equal to x, provided that I pulled x out of an urn. So the distribution is that there is 0 chance of the species going extinct before now (which makes sense!) because the ball did not come from an urn with fewer than x balls, and equal (and, as it turns out, infinitely small, if there is no upper bound to time/population) chances for any two times in the future, represented by any two urns with more than x balls.

Then, of course, even that reasoning is overly simplistic, because it doesn't apply any weighting to the urns based on social factors, when our star is due to blow up, science, etc. But it's the correct logical conclusion to an unbiased version of the urn example.