Point taken - in some cases, the significance of the gaps is more evident to the outside view.
In that case, we can replace "point out where in their argument they went wrong" with "point out where our underlying value judgments seem to diverge."
If they then try to argue that your values are wrong and theirs are right, either you have to move the discussion up a meta-level or, yes, screaming.
An application of this hierarchy:
Jack the Scarecrow. My crystal healing pills will give you eternal life. For $50.00 each, you need never die, suckers.
DH0: "I'm not interested for myself, but can I buy you a border collie and give her some? If you're going to live forever, you're going to need a smart friend to make the really tricky decisions."
DH1: What, exactly, is your profit margin on these crystal healing pills? If we don't live forever, would you still make money off of them?
DH2: Any post that ends in the word "suckers" directed at the readers is difficult to read charitably.
DH3: > My crystal healing pills will give you eternal life.
According to this hierarchy, D3 is arguing on a higher level than DH0, DH1, and DH2. And, well, maybe that's correct. "Higher" doesn't necessarily mean "more subtle." But in the absence of this hierarchy, I'd have been tempted to order the arguments from low- to high-level as 0 < 3 < 2 < 1.
(EDITED to remove potential applause light; EDITED again to better hit the idea of DH2)
Why are my opponents ignoring what I say because I said it angrily, or sadly, or confrontationally, or in passing, or whatever?
The way you say something may signal that you are trying to diminish their status. If you say it with a sufficiently negative tone, it may even be taken as a signal (a generally reliable signal) that you care more about diminishing their status than about having a truth-seeking discussion.
In other words, what wedrifid said, but less simply and more explicitly.
I think that in some contexts, like arguing over mathematical proofs (as orthonormal noted), spending a little time arguing with yourself to bring out X'Y'Z' is polite and a sign of good faith. In other cases, I'd rather just trot out A'B'C' early on, as long as it doesn't require too much effort, and deal with both arguments at once without ever explicitly raising X'Y'Z'.
I was aware of the genre it spoofed, but I didn't know that it was so specifically targeted. I'm tempted to try to find that made-for-TV movie and watch clips just to increase my appreciation of Airplane!
In this case, I'd even drop my initial thoughts about rudeness. If you can prove that somebody's gone down mathematical blind alley, it's downright polite to do so, since there's no ambiguity about the relevance of the steel man here.
Ideally, a reasonable counterargument that applies to the strong form will also apply to the weak form without significant editing. If the person one was arguing with would have been receptive to DH7 in the first place, that alone should stop them from making the strong form argument - the countering evidence has already been provided.
Where this fails... well, I said "at first" in my thread-starter for a reason.
I don't ALWAYS have low confidence in the other arguer's ability to tolerate a steel man version of their own argument. I do have low confidence in the ability of most people, especially me, to decide what constitutes a non-gratuitous steel man. I have an unfortunate, but understandable, bias in favor of my own creations, and I suspect that this bias is widely shared.
I'd say that the incoherent speaker is arguing at DH(-1). DH0 would be an improvement. You would be counterarguing at DH(No) - argument by pointing out conversational emptiness.
(edited to clarify that it is the person who makes the incoherent argument who is arguing badly, and the person arguing against that who is doing something entirely outside the hierarchy.
Other DH(No) arguments-that-are-about-non-argument include "We aren't actually arguing about the same thing" and "let me take some time to do more reading before I reply.")