Unfortunately, centrists do not have a great long-term track record of opposing genocide. Wish they did. But when you've got a mass of anti-war marchers in the streets or protests outside embassies or pickets outside Presidential ranches or in front of party convention halls, the centrist response tends to be "That's not very civil! Tut-tut!" right before police wade in and start cracking heads.
I really want to know where you get the idea that the left wants genocide, especially over a difference of opinion. It's so antithetical to my values that it's downright funny.