>> that shortly after the detector came online we just happened to observe the exact moment a billion years ago that two black holes collided?

Counterintuitive, but yes. Because it happened billions of years ago, it happened a long long way away. The sphere of objects billions of years away/ago is far larger than those closer to us. So such a detector should be detecting exponentially more very old objects than new ones. Given the rarity, I would expect nearly all detected events to have happened long long ago in galaxies far far away.

Also, models point to such events being more common in the distant past where there were more black holes (primordials) floating around than there are now.