I think that the main reason of fighting Muslim countries is power. they perused realistic approach in the own relation and using Sunni and Shiites just is a tools for getting power.
Agree with politics. If you look at immigrant Muslims societies in Western countries, a madhhab conflict is rare between people. However, the conflict is intense in the Middle East especially between the states. Now, it seems like a power play, rather than a historical conflict.
If you are interested in historical debate, i'd suggests some reading on Ali, Battle of Karbala, 12 Imams.
Maybe inventing a cure for greediness would prove feasible.