From wikipedia https://en.wikipedia.org/wiki/Bias_of_an_estimator :
because a biased estimator gives a lower value of some loss function (particularly mean squared error) compared with unbiased estimators (notably in shrinkage estimators)
Further evidence is provided in the linked wiki article on shrinkage estimators https://en.wikipedia.org/wiki/Shrinkage_estimator:
A well-known example arises in the estimation of the population variance by sample variance. For a sample size of n, the use of a divisor n − 1 in the usual formula (Bessel's correction) gives an unbiased estimator, while other divisors have lower MSE, at the expense of bias. The optimal choice of divisor (weighting of shrinkage) depends on the excess kurtosis of the population, as discussed at mean squared error: variance, but one can always do better (in terms of MSE) than the unbiased estimator; for the normal distribution a divisor of n + 1 gives one which has the minimum mean square error.
An insight into the tradeoffs of minimizing MSE vs retaining the properites of unbiased estimation and consistency would be helpful. Some practical insights and/or examples into when a biased estimator would be preferred would also be appreciated.