[...]
We've probably got our wires crossed somewhere, because I can't reconcile you agreeing with post #2 (in that thread) with you taking issue with my post (which is just a high-effort restatement of the same idea: that you shouldn't expect the shadow to get bigger).
I think we're taking different views on the question. I like your framing of the problem in terms of heat transfer, but my post was only concerned with (idealized) shadow length. I think it's one of those occasions where you're concerned with what the OP was
really asking, whereas I'm concerned with what the OP was
actually asking.
My post started with a snip from the OP that amounts to (paraphrasing): "If I lift a 10 ft by 10 ft tarp, 10 ft into the air (at noon) how big would the shadow be?". And the rest of that post is basically justification for my answer: ~10.0000000002 ft (squared).
Viewed through that lens, I think it's sensible to ignore the size of the sun, because it plays no significant role in understanding why the shadow size increases by such a small amount (in fact, it's likely to confuse the issue, with edge softening that makes gauging "size" more difficult).
The fact that light arriving from more than one position "eats" into the shadow cast by light arriving from a chosen position doesn't affect the underlying relationship [1] that predicts how shadow size varies with distance (to see why this is true, you can always decompose any light source that is not itself a point source — all realistic lights, then — into a swarm of new point sources, and the size of the individual shadows cast by each of these new lights will be governed by the same relationship).
[1] Which (to be clear) is more general than the one I special-cased for an overhead sun.