why might the american public be wary of a central government having too much power over local affairs and problems

I would say that mainly because of three reasons:

Federalism: The United States is a federal republic, where each individual state has power over local affairs. The United States is not a centralized country like France where officials from Paris can take decisions in regards to every single aspect in every single region. Fear of Tyranny or Dictatorship: many americans are afraid that a too powerful central goverment might evolve into a dictatorship, because it would acquire more and more power with time. Fear of centralized incompetence: many americans believe that central governments are more incompetent than local governments. People think that policymakers in Washington do not have enough information about, say, Idaho, to do a good job in dealing with problems in that state. They feel that local Idaho politicians can do a better job because they live and work there.