How does resolving the problem between the Sunnis and Shiites benefit the US?

Answer 1

Shape of Resolution
First it should be noted that the United States cannot resolve the theological schism and go back to one single Islamic faith; this is impossible. There are irreconcilable questions of belief that separate Sunnis and Shiites such as: (1) Who was the first legitimate Caliph: Abu Bakr or Ali? (2) Is Jihad a Pillar of Islam? (3) Do the Infallible Imams carry jurisprudential weight? (4) Are shellfish halal or haram? etc. In order for the schism to undo itself, either the 250 million Shiites or the 1.3 billion Sunnis would have to concede to the answers opposite of what they believe on these and similar questions.

However, what the US should strive to do is not to strive for union but strong and cordial relations between Sunni and Shiite communities, not unlike what the Catholic and Orthodox Churches retain now 1000 years after their fractuous schism. They recognize each others' differences but collaborate on mutual goals. Of course, the benefit to promoting this in the Middle East for Middle Easterners is obvious. If sectarianism can be minimized, it will make life safer and more free for numerous Middle Easterners and South Asians. Democracy will have a chance to ennoble and improve the lives of numerous Middle Easterners and South Asians.

Benefits that accrue to the United States
While the benefits that accrue to Middle Easterners and South Asian from resolving sectarianism between Muslims in those region are relatively clear, the benefits that accrue to the United States are less obvious, but no less important. They include:

Trade Stability: One of the major reasons for fluctuations in the American economy has been due to the price of petroleum, which is directly correlated with instability in the Middle East. Helping to stabilize the Middle East will thus correlate with stability in the US economy.

Bulwarks Against Islamism: Islamism and other theocratic philosophies with which the United States cannot reasonably treat are opposed by the same individuals who are promoting reconciliation between the Islamic Sects. Empowering those individuals who wish to reconcile Sunnis and Shiites and other non-sectarians will form a bulwark against the rise of the Islamists. Combating the rise of Islamism is directly in the US interest not only because the United States cannot have stable and beneficial relations with Islamist regimes (Taliban Afghanistan, the Islamic Republic of Iran, Sudan, al-Shabaab in Somalia, and ISIL), but because these regimes support and protect terrorist groups like al-Qaeda which have violent designs on the US and its citizens.

Regional Allies: There is no better way to forge allies in the Middle East than by being a peacemaker. The United States' Islamic Allies in the region are problematic because most of them practice open discrimination and do not endorse democratic values. Helping to resolve the sectarian tensions will create democratic allies, like Turkey and Israel that the United States can have strong and beneficial relations with.

American Redemption: While public opinion of the US in most Middle Eastern and South Asian Islamic circles will never be very positive, by helping and promoting peacemakers in the Islamic communities in the Middle East, some ground can be made in improving the US image in the Islamic World. Kosovars and Kurds have an incredibly positive view of the United States for having helped to end the slaughter of their peoples by sectarian thugs. There may yet be hope for US Redemption in Arab, Iranian, Afghan, and Pakistani eyes.