I have used and currently recommend this method with all kinds of conflict. Restate the disagreement in terms the other party will agree with. When you have done this, to their satisfaction, additional conversations can happen.
From the description of what they do, it seems like something that you need to have some personal investment in making work - family, religious community, etc. It's a bit easier to come to an understanding of someone on "the other side" when you're motivated to preserve your relationship with them.
Unfortunately, there's also a huge incentive for various parties to maintain/ramp up the conflict. Rage has been a pretty good political motivator over the past few years in the US, and it makes for compelling infotainment on cable news networks, whose ratings are significantly driven by the emotional button pushing opinion shows - Tucker Carlson, Rachel Maddow, etc.
But maybe the only progress to be made is in small personal situations that change people's attitudes in other settings.
I agree completely that it is really difficult if not impossible to scale. But if you consider that sometimes it takes just a few key community leaders to get together and hash things out, that inter-leader level of dialogue can really scale.
