Why might some people believe that the Fed should not be an independent entity but should coordinate with the government?
The reasons for giving the elected branches of government more control over the Fed are more political than they are economic. From an economic point of view, it is better to have the Fed remain independent so that it can do what is best for the economy in the long run instead of what is better in the short term. From a political point of view, though, there are problems with this.
The real issue here is whether the government is to be controlled by voters. There are many things that technocrats could probably do better than elected politicians. The problem is that the more jobs that we give to technocrats, the less democratic our system is. The Fed is entrusted with making and implementing monetary policy. This means that they can have a tremendous impact on the health of our economy. It is somewhat antidemocratic to have a system in which an unelected body can have so much control over our economy.
If we wanted everything to run just the way experts think it should, we could simply do away with democracy and have a government of unelected experts. This would, however, be antidemocratic. Many people feel that the Fed should be controlled more by the elected government officials so that it would be more in line with our democratic values.