Mean Squared Error as quantifier of the Bias-Variance tradeoff

Question

I have acquired the impression that many of the people doing statistical work, will prefer a biased estimator $\hat b$ to an unbiased one $\hat \beta$, if the former has lower Mean Squared Error.
This appears to be a way to quantify the "bias-variance trade off" (excerpt from the CV tag of the same name):

...In predictive modeling, unbiased models can have higher variance, & thus be less accurate. Modelers may prefer some bias to maximize accuracy...

...since, in order to have lower MSE, it means that the variance is not just lower, but lower below a specific threshold,

$$MSE(\hat b) < MSE(\hat \beta) \implies \text{Var}(\hat b) < \text{Var}(\hat \beta) - \text{Bias}^2(\hat b)$$

My request is for an elaboration of the argument as to why MSE is a reasonable quantifier of the "bias-variance tradeoff", and related references hopefully.

...or against, proposing some other measure.

I link here two threads, which provide arguments in favor of using MSE or more generally "squared differences" as "loss functions",

Why is the squared difference so commonly used?

and one that I have also contributed,

What makes mean square error so good?

but I stress that I am mostly interested in whether there is some argument about why MSE is a good quantifier of the bias-variance trade off.

Answer 1

I think you're thinking of this in the wrong way. It isn't that MSE quantifies bias-variance tradeoff. It's that people care about MSE, and MSE can be expressed as in terms of bias and variance. Because of that, people are willing to accept a bit of bias in exchange for a large reduction in variance, since that reduces the quantity they care about (MSE).