So it seems you need at least some guess of what the distribution might look like to quantify the likelihood of guessing correctly that you're in a high population country. But your guessed distribution doesn't have to be perfect. If you knew the exact distribution, you could compute the exact chance that a random individual is from a high pop country (just take the integral of the distribution). But if you don't know the exact distribution, you can abstract one level up, try to reason about the range of possible distributions there could be, and come up with a probability based on your more limited knowledge. At least, that's what I understood from the article.
For the actual calculations, the author links to them in the article, but I haven't looked at them myself.
Instead, the whole thing depends critically on other arguments about the possible distributions, which are much weaker and than the initial ironclad statistical argument.
If you chose Iceland and used this logic to conclude you're in a high pop country, you'd be wrong. But of course, most people don't live in Iceland. You probably don't live in Iceland. If you pick a random person on Earth, odds are very good that they live in a high pop country like France, the US, China, India, etc.
If you pick a random country, there's a 50% chance you'll pick a country with a lower population than the median country. But not if you pick a random human from Earth.