What's the goal of this article?

There exists a problem in real life that you can solve in the simple case, and invoke a theorem in the general case.

Sure, it's unintuitive that I shouldn't go all in on the smallest variance choice. That's a great start. But, learning the formula and a proof doesn't update that bad intuition. How can I get a generalizable feel for these types of problems? Is there a more satisfying "why" than "because the math works out"? Does anyone else find it much easier to criticize others than themselves and wants to proofread my next blog post?

Here's my intuition: you can reduce the variance of a measurement by averaging multiple independent measurements. That's because when they're independent, the worst-case scenario of the errors all lining up is pretty unlikely. This is a slightly different situation, because the random variable aren't necessarily measurements of a single quantity, but otherwise it's pretty similar, and the intuition about multiple independent errors being unlikely to all line up still applies.

Once you have that intuition, the math just tells you what the optimal mix is, if you want to minimize the variance.