I have found this very interesting paper by Bruce Hansen https://www.ssc.wisc.edu/~bhansen/papers/gauss.pdf (conditionally accepted at Econometrica). This is actually a great result that should change the way we teach OLS. The proof of the Gauss-Markov theorem was a good exercise but the limitation to linear estimators is difficult to justify.

Unfortunately, I do not understand his proof even for the mean. Hansen shows that the variance of the sample mean is equal to the Cramer-Rao lower bound for a particular parametric model. Why does that guarantee that this lower bound holds for all models satisfying the assumption?

I also have another question: Does any unbiased nonlinear estimator of the population mean even exit?