Actually I've interpreted that phrase differently....
Both sides know they will disagree endlessly, and that no consensus can be made.
So it's a more polite way of saying.... "We wil just argue forever, so any further discussion/debate is a waste of time. This conversation is done. Bye."
In this context, the phrase can be a useful (and tactful) way to end a heated discussion.
Bravo, and very well said; and yes, I agree entirely with your interpretation, for this is how - or, this is the context where - I will use that expression.
It can be a useful and tactful way with which to end a disagreement, yes, but it also signals - publicly - that you will not alter, or change, your stance, - that you stand over your argument - but neither do you expect that the position of the person with whom you are in disagreement will change.
As a woman, I find this one very useful, because, sometimes in arguments, or disagreements, men (among others) will expect you to give way almost automatically, and this allows you to make the point - clearly, but politely - that while you are withdrawing from the debate, your position remains unchanged.