First of all, I am not sure what “story line” you are asking about. Secondly, a republic and a unitary government are not opposites. It is entirely possible to have a republic that is also a unitary government. Therefore, it is not possible to say why the colonists wanted a republic instead of a unitary government. I can, however, tell you why the American colonists wanted a republic and why they did not want a unitary government.
A unitary government is one in which the central government has all the power and the lower levels of government only have whatever powers the central government gives them. This would be like a system where the US government had all of the power and the states could only do whatever the national government let them do. The British system was unitary. The government in London controlled everything and the colonies only had whatever powers that government allowed them to have. The colonists did not want this kind of government again because they felt that the central government was too far away from them and did not understand or care about the issues they faced. They wanted a system of government where the lower levels had more power.
The colonists wanted a republic rather than a monarchy because they wanted the people to be in control. They believed, as Thomas Jefferson said in the Declaration of Independence, that the government got its “just powers from the consent of the governed.” They did not think that a government that was controlled by a king could be just because it would not be based on popular sovereignty (the rule of the people). They wanted a system in which the government had to answer to the people.