I actually am thinking statistically; if you want to be formal, you're arguing that the probability of von Neumann beacons existing (as opposed to von Neumann probes without beacons) is non-zero and that enough civilizations exist that at least one would deploy them. I'm arguing that:

a) advanced civilizations may realize that attaching beacons to vN probes is a stupid idea and that the probability of one of them doing it may well be 0, especially if they can detect us and contact us at will without need for a clumsy beacon

b) even if the probability is non-0, the only thing suggesting that in absentia a great filter there would be 'untold billions' of hyper advanced civs is your flowery rhetroic

c) the decisions made by hyper-advanced civilizations probably will not be independent (which kills your attempt at a law of large numbers) -- for example, a civilizations may well just share a single vN probe network or that civilizations may find and piggyback off of existing vN probe networks. After all, this is what humanity does -- or are you every day baffled by the statistical mystery that there aren't a dozen different parallel fiber optic lines running to your house, and that inventors don't repeatedly re-invent the lightbulb?

Oh yeah, and then there's the obvious matter that civilizations may well not permit others to infringe on their vN network's territory, destroying any obvious vN probes approaching their lands.

Also, just for reality check, consider that your argument can be applied to a wide range of absurd scenarios. What's the probability that an alien civ will consider all other life forms to be a threat and equip their vN probes with extermination devices? >0 * n\to \infty civilizations = we're dead. Oh wait, but what's the probability that another such civ would defend us with their vN probes! Non-zero, great, n \to \infty civilizations * non-zero probability of extermination probes * non-zero probability of defense probes = 100% chance of there being a probe war over head. Oh wait, and now what's the probability that some civ thinks it'd be amusing to send out vN probes that make crop circles? Probably non-zero. * untold billions of civilizations = there are crop circle probes fucking with farmers as we speak.

Or maybe, civs don't send out von Neumann probes with beacons on them because they'd prefer to contact us at their whim, rather than be found at ours? Or more maybe still, perhaps it's more likely than not that only one or a few vN probe networks will exist in any point of space for reasons relating to advanced civs' decisions which, undoubtedly, will condition on the decisions of all other such civs!

tl;dr "if it's possible, it's happened in the limit" only works under specific assumptions that need to be defended with something better than flowery rhetoric and a sci-fi fueled imagination.