I do not think that the war in Iraq would of concluded faster if the troops reached out to survivors of the Kurdish genocide. I think it might have of made some difference but overall I think that our troops would still be over there and still be doing the same thing.
These wars would not conclude sooner because the Kurdish genocide was only one of the events that led to these wars. And the Kurdish people might be happier with this reach-out, but these wars have also created many issues on top of the Kurdish one that would all have to be addressed seperately -- for example the fight for oil, understanding who harbors WMD, and religious wars amongst other citizens within these countries.
No, the war in Iraq and Afghanistan would not conclude sooner if American forces were to specifically reach out to survivors of the Kurdish genocide, because that would create internal discontent, and it would ultimately take longer for the war to end. American should not convince the locals that the military effort in those countries is about any particular ethnic subgroup, because that will create ongoing loyalty issues.
The Iraq War wouldn't have ended sooner because Kurds were outgunned by Iraqi forces to begin with. The war in Afghanistan wouldn't have ended sooner because the Taliban had control and Kurds weren't as prominent in Afghanistan as they are in Iraq. American and coalition forces were the only ones who could invade both countries and be successful. Inviting other forces to help within those countries would have only caused more problems and extended the conflicts.
This has little relation to the reasons why conflict still exists in the region. While obviousy attempts at making peace are a major part of how to steer away from war, this isn't about any one group of people at this point. Everybody in the region is mad about something and it's far too much of a mess to repair it by addressing any singular issue.