The system was set up in this way to prevent economic decisions from being made on a political basis. The Federal Reserve (the Fed) is supposed to be a group of economic experts who will make their decisions based on what is best for the economy. They will not base their decisions on whatever is most likely to be beneficial to a certain party’s goals in the upcoming election.
Monetary policy is supposed to be something that can be determined technocratically. That is, it should be determined scientifically, based on the needs of the economy. The proper monetary policy is, in theory, not a matter of opinion. Therefore, monetary policy should not be determined by politicians. It should be determined by experts who will only pay attention to what the statistics say the monetary policy should be.
If we set up a system where the president (or Congress) had control over monetary policy, that policy would be set based on political needs. The president would always try to stimulate the economy so that things would be going really well at election time. Presidents would have an incentive to do what was good for them in the short term even if that happened to harm the economy in the long term.