Central bank interest rates don't exist to 'control prices' per se. The state would like to create new money supply to make the economy move faster (e.g. maintain high employment in slow business periods). This would be to the disadvantage of the banks (their loans to debtors would be devalued). Hence a compromise was created to make the money supply more elastic without risking hyperinflation - banks may create new money supply for this purpose but must pay for it at the interest rate. Inflation is key a part of the feedback loop, to limit money creation to some supposedly optimal target range.

Surprisingly, this system was created by bankers and politicians, rather than economists /s

There are also many sources of 'cost of living' which are not related to money supply and therefore not directly related to monetary policy.