I don't think that the rarity of starvation in modern times is due to central banks, but instead due to advances in farming that allows a very small percentage of people to produce enough food for everyone.
Physicians don't think they can manage and control the billions of cells in our bodies; we still don't understand how all of our cells even work. Physicians apply tested practices to individual people and hope that it addresses whatever medical problem is being observed. They understand that treatments are not one-size-fits-all and that mistakes can kill people. They also understand that our bodies self regulate and they are only trying to address a specific imbalance in their specific patient, not something that applies to an entire population.
Central banks think they can turn a couple of knobs (currency supply and interest rates) and control an entire economy made up of millions of people. Using the physician example, that's like thinking that the only two treatments needed for any medical condition are adjustments to our blood level and body temperature.
Monetary policy is a newer invention in the history of human civilization and even then, it wasn't always applied so universally. Monetary policy can only exist with central banks.