Most of the Civil Rights Era concessions were the result of non-centrists organizing non-centrists and making so much hell and disruption that the centrist was, time after time, forced to concede, until finally the centrists began throwing the secret police at organizers. Appealing to the moderate's underdeveloped morality works for the right (because the US is a right-wing country) but will always comes across as inauthentic when from the left.
The goal of any organizer is to shift the gestalt, not surrender to it. Surrender to the moderates is what killed Reconstruction in this country, killed any chance of a functioning, modern healthcare system, and is the number one thing holding back the abolition of chattel slavery in the American prison system.