I like to think of it like finding a dozen cats in a forest as large as the surface area of the earth. Scatter those cats in there randomly and then take a few steps into the forest. Will you be shocked if you don't find them after a few steps in? Take five or six more? No cats? Hmmmm...must be a paradox. We know they are in there, so why aren't we finding them? Scale this up by a magnitude of a few billion.
, you forgot to give these cats an evolutionary advantage that ensures their survival and prosperity (equivalent of intelligence/technology) and to wait a few thousand years. Fulfill those two requirements and, when you walk into the forest, you'll find the cats everywhere.
For a more detailed explanation, read my previous post from this thread.
Actually there is a flaw in my analogy. We do have an idea of how big space is, but we have no idea how widespread alien life is should that life exist. Further, even if a small percent of life has spread out from their home planet to other places, we do not know how prolific they are. Do they really spread quickly from galaxy to galaxy, or just populate a very small number (say a few thousand) planets in the scheme of things. How many of these civilizations may populate even a handful of planets, much less a thousand. How many go on to populate multiple galaxies?
Even if life is fairly widespread, given the largeness of the universe, I do not think comparing 12 cats living on a land mass the size of the earth is any kind of obvious understatement to the population density of the universe. It seems more likely that it is an overstatement.