Your Earth-volume : lunar-orbit-volume ratio works for the likelihood of finding the asteroid inside or outside of the Earth at any given moment.

But we can assume the object has an entire path of moments passing straight through on [basically] a line. An asteroid “looking ahead” directly at the round perimeter of the Earth might briefly occupy some point “in front” of the Earth before collision… but that’s still a collision path. And it’s never gonna get to any points on the other side. What it “sees” just is a flat disc in a flat disc. It’s either heading through the empty part or not.