I wonder if this minimum variance approach of averaging the measurements agrees with the estimate of the expected value we'd get from a Bayesian approach, at least in a simple scenario, say a uniform prior over the thing we're measuring and assume that our two measuring devices have unbiased errors described by normal distributions.

At least in the mathematically simpler scenario of a gaussian prior and gaussian observations, the posterior mean is computed by weighing by the the inverses of variances (aka precisions) just like this.

https://en.wikipedia.org/wiki/Conjugate_prior