Indeed, the unbiasedness condition for an estimator says nothing about its quality relative to other estimators. Instead, it requires that you cannot change the

Indeed, the unbiasedness condition for an estimator says nothing about its quality relative to other estimators. Instead, it requires that you cannot change the target function γ without the expected loss becoming worse. The paper I referenced even includes a theorem saying that, as long as the value to be estimated can not be exactly deduced from the observation, no estimator is both Bayes-optimal and unbiased. However for many observations, the Bayes-optimal estimator becomes asymptotically unbiased.