Before I respond to that, which I'm more than willing to do (I'm enjoying this), is it out of place to suggest you first identify a flaw in the formal argument I laid out, not a link to an intuition that could or could not be misleading? All the interesting problems in statistics are subtle, and it's burdensome to some extent to lay out this math to chase down each intuition with the amount of rigor I'm trying to provide. The debate doesn't progress if one side is conversing in math and the other is conversing in intuitions.

Not to be cantankerous, but I think that's the only way this will get figured out.