Imagine a series of one-shot random meetings between central banks and publics that know nothing about each other. The problem results from the central bankers' preferences combined with policy discretion- but leads to results even the central bankers don't like. H. Time consistency often used to explain failure of discretionary policy in the '70's - and around the world in high inflation countries today. It is then possible to solve for the critical value of b . If b is large enough, then a reputational solution is possible - though hardly necessary.
Data actually seem fairly consistent with simple model without supply shocks.
If more CB independence eventually makes real performance worse, then it seems hard to find examples of countries that have that much CB independence.