As with so many ideas in American politics, the use of moral outrage has ping-ponged back and and forth across the political spectrum -- everyone uses it, and people have for a long time. The civil rights era: both sides used moral outrage. Prohibition/the temperance movement. The suffragettes being force-fed during their hunger strike in 1909, and the moral outrage about corrupting women and families and degrading America by allowing women to vote.
More recently, social issues have been the wedge between Democrats and Republicans in the US. The Republicans leveraged abortion and contraception because the killing of innocent babies via abortion and/or contraception is the worst sin a human can commit (moral outrage). Now progressives leverage LGBQT+ rights because erasing the humanity of a living person is the worst sin a human can commit (moral outrage). Climate change has both sides too; killing the planet for the convenience of cars and plastic bags is worst; sacrificing human economic progress and security for some stupid turtles and this made-up sea level thing is the worst.
Moral outrage is just a tool. Research is accumulating on the addictiveness of self-righteousness. Moral superiority and outrage get people to do stuff (vote, give money, show up at protests, sign petitions), in a way that self-reflective questioning does not.
How could we engineer things so that "moderate compromise with room to listen to the views of others" would get people to act?
You seem to be narrowing the issue and abstracting it away from the local problems until you can easily dismiss it as a mere tool everyone has employed historically. But I don't think any recent history has had to have a perfectly "acceptable" political ideology and personal history (sometimes going back 40+ years) - that hasn't "offended" multiple extreme ends of various political camps.
...absent the Victorian age and some time after, which we all moved beyond culturally because it was backwards to force one top-down rigid "proper" way to live and speak.
Not to mention the definition of "acceptable" to these special interest camps is forever a moving goal post where anyone who isn't the perfect caricature of mainstream "acceptableness" can't have a say in any economic, cultural, political, etc discourse without being tarred, feather, and demanded to apologize for even thinking they could engage in the process.
This is a very different development than just employing moral outrage. A critical part of a healthy democratic society is accepting people have different worldviews. We're moving to a world where there is one alleged correct worldview and anything else is a showstopper.
More recently, social issues have been the wedge between Democrats and Republicans in the US. The Republicans leveraged abortion and contraception because the killing of innocent babies via abortion and/or contraception is the worst sin a human can commit (moral outrage). Now progressives leverage LGBQT+ rights because erasing the humanity of a living person is the worst sin a human can commit (moral outrage). Climate change has both sides too; killing the planet for the convenience of cars and plastic bags is worst; sacrificing human economic progress and security for some stupid turtles and this made-up sea level thing is the worst.
Moral outrage is just a tool. Research is accumulating on the addictiveness of self-righteousness. Moral superiority and outrage get people to do stuff (vote, give money, show up at protests, sign petitions), in a way that self-reflective questioning does not.
How could we engineer things so that "moderate compromise with room to listen to the views of others" would get people to act?
(Edited to add a paper on self-righteousness, with experiments! http://home.uchicago.edu/~nklein/KleinEpleyLessEvil.pdf)