(Disclaimer, stats noob here) - I thought the point was that, you have a better chance of being -overall- closer to the mean (i.e., the 3D euclidean distance between your guess and the mean would be the smallest, on average), even though you may not necessarily have improved your odds of guessing any of the single individual means.

So it's not that "you get a better estimate of the US wheat yield by estimating also the number of Wimbledon spectators and the weight of a candy bar in a shop", it's simply that you get a better estimate for the combined vector of the three means. (Which, in this case, the vector of the three means is probably meaningless, since the three data sets are entirely unrelated. But we could also imagine scenarios where that vector is meaningful.)

Am I misunderstanding something?