and that their choice of weights minimizes the error

The author has selected a weighted average such that if we treat that weighted average as a random variable, its standard deviation is minimized. But if we just want a random variable whose standard deviation is minimized, we could have a distribution which assigns 100% credence to the number 0 and be done with it. In other words, my question is whether the procedure in this post can be put on a firmer philosophical foundation. Or whether there is some alternate derivation/problem formulation (e.g. a mixture model) that gets us the same formula.

Another way of getting at the same idea: There are potentially other procedures one could use to create a "blended estimate", for example, you could find the point such that the product of the likelihoods of the two distributions is maximized, or take a weighted average of the two estimates using e.g. (1/sigma) as the weight of each estimate. Is there a justification for using this particular loss function, of finding a random variable constructed via weighted average whose variance is minimized? It seems to me that this procedure is a little weird because it's the random variable that corresponds to the person's age that we really care about. We should be looking "upstream" of the estimates, but instead we're going "downstream" (where up/down stream roughly correspond to the direction of arrows in a Bayesian network).