To get things clear, Sufism is not one sect. It is a belief in a mystical and more figurative version of Islam. It is simply an understanding of Islam. The real root of problems in the Middle East is Salafism/Wahhabism. This is the Saudi/ISIS/Al-Qaeda/Taliban version of Islam, a very literal and brute version. Sufism is practiced in Iran and many Shiite areas. Do you ever see them blowing themselves up? Trust me, I know firsthand. It would.
It doesn't matter what religion people adopt in the Middle East or elsewhere, a single religion is not the answer. Rather, people need to learn to get along not only with the people they agree with, but also the people they disagree with. Widespread adherence to Sufism may not solve the problem, but respecting all religions is.
I believe at this point the Middle East is way to fractured with many militant religious groups of many different religions, so even if you got many people to adhere to one religion those that are the most extremist would still cause the same amount of trouble that they do today, so I don't think it would do much good.
I do not think a more general of acceptance of Sufism in the Middle East will reduce the number of conflicts. I think that the groups of people that have problems there will always have issues regardless of a change of ideology or not. I think the Middle East needs more than just more philosophy.
Widespread adherence to Sufism would not reduce the number of conflicts in the Middle East. Sufism is still an adherence to Islam, and much of the conflicts in the Middle East are between Muslim countries and Israel, or between Muslims and Jews. So Sufism would not reduce the number of conflicts in the Middle East.
The main difference in the different sects of Islam are the structure of the church. The rules of day to day life remain the same. The principles or rules of Islam are the same. The only thing that's different is who is allowed to tell you. Even if Sufism were to expand to the majority, the actual conflicts would not stop. The conflicts are typically over land. Religion is just an excuse for recruitment. They will find new reasons to kill each other. The result will always be either power or land though.