"Noerdlinger counters by saying that the 'proto-moon' did have an iron-core, but then Earth's gravity pulled
the iron-core to Earth, leaving behind a rocky shell which reassembled into our iron-free [iron-defficient?]
Moon."

But earth doesn't pull the moon toward it, now, nor should it have done so in the past. And dense iron core wouldn't be drawn towards the earth for any reason more so than lighter material. Nor do I see any reason dense material would form faster in space environment. I see why massive objects would form faster, but abundance of whatever makes up the gas cloud will play significant role in early composition of proto-moon.