Do religions solve political problems or add to them?

It is far too complex a question for there to be a simple this-or-that answer. It is clear from history that religion has played a key role in many major political disagreements including wars, but it is also clear that some religions have at times been strong forces for peace. Three of the great world religions, Judaism, Christianity and Islam, have each been on both sides of this argument at various times. Other world religions like Hinduism and Buddhism are fundamentally more oriented toward peace.

It may be more useful to explore the relationship between governments that do not allow the establishment of state religion and those that do. There is no implication that one way is necessarily preferable as long as the general principle of no compulsion in religion is adhered to where state religion exists. The US does not allow the establishment of religion while the UK has established the Anglican Church as the state religion with the monarch as its temporal head. The UK has broad and powerful protections for those who wish to practice some other religion or no religion at all.

There is also a difference between a government establishing a state religion, and a government that for all intents and purposes sees no difference at all between civil government and the practice of and adherence to a specific religion. It may be that maintaining freedom from compulsion in religion is more difficult to protect in some of these approaches to religion and government.

Everyone should consider that religion in principle is not the issue. The problem is adherents of religion that take on a hyper-evangelistic or militant mission to literally destroy all systems and peoples that do not hold to one specific set of religious beliefs and practices.