Hmm. That conclusion is not where I thought you were going with the war analogy.

Suppose that the United States finds itself at war with, say, Iceland. At the height of the conflict, the two sides sit down at a negotiating table, whereupon the American diplomat says to his Icelandic counterpart: “You guys know you can’t win this one, right?”

In this scenario, are we to suppose that U.S. leaders are worried about losing the war… to Iceland? Are strategists at the Pentagon frantically drawing up plans for resisting an Icelandic invasion of Maine (with the rest of New England—according to their projections—soon to follow)?

Seems unlikely. Rather, the message here is more like: “You can’t win, and we both know it. The only uncertainty in the outcome is how much money you can make us spend on beating you. And if you make us fight you to the end, we’ll make you regret it. There won’t be a hut left standing in your whole country, when we’re done with you. So give up now, and we’ll both come out on top, compared to that counterfactual scenario. We don’t have to fight; and it’s better for both of us if we don’t. But if you insist on fighting, you definitely can’t win… and losing will be much worse for you than not fighting at all.”

How does this map to the argument scenario? Well, consider two sides, A and B (where B is just “everyone who isn’t A, and cares about the question at all”); and representatives of each side, respectively Alice and Bob. Alice makes some argument, aimed at persuading members of B to the A position. Bob (who is a B) says: “You’ll never persuade people like that.”

It seems like one obvious expansion is:

“You won’t persuade any Bs like that, and we both know it. The only uncertainty in the outcome is how much effort you can make me and other Bs expend on countering your arguments. And if you make us deconstruct your arguments in detail, we’ll make you regret it. You’ll look like an idiot, a lunatic, an ignoramus, when we’re done with you. So give up now, and we’ll both come out on top, compared to that counterfactual scenario. We don’t have to have this argument; and it’s better for both of us if we didn’t. But if you insist on arguing, you definitely can’t win… and losing will be much worse for you than not trying to argue at all.”

When a person is acting on behalf of a group, persuading the person doesn't affect their stance, you'd need to either persuade the group as a whole or convince that person to do something other than acting on behalf of the group. So in this case the metaphorical hard-to-reach boss is socially real, not centrally made up.

To give a concrete example where I said this:

My father in law wrote a book about the issue of divorce refusers in Judaism, and the apathy of the UK batei din towards the victims of this abuse.

In it he argued that if the rabbanim had been willing to, they could easily find a basis on which to unilaterally dissolve such a marriage.

I personally think that's true both as a question of historical fact (rabbis have found halachic justification to do just about anything when they really wanted to), and also from the perspective of his modern orthodox philosophy which recognises that as a legitimate flexibility important for Judaism's survival.

But having grown up Hareidi, I had a better insight into the perspective of the extremely conservative ultra orthodox Jews who run the batei din. And from their perspective, Judaism is basically unchanging, and given that the authors of the Talmud clearly tried to find a solution to the issue of agunot, and could not, it's a closed case. There's nothing to do here.

So I explained to him that whilst I might agree with him, all his arguments will do nothing to persuade his target audience, and would get him branded as at best naive, and at worse a heretic. He would do better off by completely removing that line of argument from his book, and focussing purely on the legal issues, on which he is legitimately an expert. "Sure, you might be right, but you'll never persuade people like that!"

And that's what explains the attractiveness of the appeal-to-persuading-third-parties. What "You'll never persuade people like that" really means is, "You are starting to persuade me against my will, and I'm laundering my cognitive dissonance by asserting that you actually need to persuade someone else who isn't here."

Big if true. Going to look out for this in future conversations. 

This, for me, usually is noticeable as an actual pivot.  "sure, that'd be nice, but it'll never happen".  There is now sufficient agreement about the main topic, and the person is now talking about why they're not working toward it, or why it doesn't matter that you have agreement in the first place.  Or they disagree, and are jumping ahead to "how can we find out?" and pointing out that you can't convince others, so there's no way to test it.

The counter-question is "why do you object to this pivot"?  Is there something remaining that you don't understand about your discussion partner's view of the nominal proposal?  Can you just ask directly "but you agree that it would be better if this were implemented" or whatever, without conceding whether there is an actual path to implementation?

[ note: this probably varies pretty widely with context and group norms.  If my explanations don't make sense, it's worth picking a few specific examples to explore how they differ. ]

I tell people that their arguments are not persuading to third parties when they insult me, because I genuinely hope that they will change the way they argue, and also because I think it's generally true that insulting anyone does not tend to convince them. I'm personally not interested in being insulted, but I am interested in people arguing for their positions well, since if they are right, I'm interested in being persuaded.

If someone said "you'll never persuade people like that" to me I'd probably just ask them what's arbitrary about my position. If it's arbitrary then they may have a point. If it's not arbitrary then people will in fact be persuaded.

Unless the belief you're arguing for is false. You checked that beforehand, right??

On further thought, I really need a few examples to understand the context of when this is happening to you.  For me, it ONLY occurs in complicated enough issues that "true or false" is very hard to define or test rigorously, and it's extremely likely that something like "directionally true" or "mostly correct" or even "preferable (to me, and I think many others) equilibrium" would be the right goalposts.  

If it's something you actually checked beforehand, you should just share that check and be done with it.  No discussion or debate required.

The bargaining model of war attempts to explain why wars are fought—and not fought; even the bitterest enemies often prefer to grudgingly make peace with each other rather than continue to fight.

However, conversely, sometimes otherwise friendly neighbours will attempt to destroy each other because they can't resolve trivial matters such as the height of a fence.