[mitthrowaway2]

Exactly this. From this perspective, the CLT then can be restated as: "it's interesting that when you add up a sufficiently large number of independent random variables, then even if you have a lot of specific detailed knowledge about each of those variables, in the end all you know about their sum is its mean and variation. But at least you do reliably know that much."

[D-Machine]

Came here basically looking to see this explanation. Normal dist is [approximately] common when summing lots of things we don't understand, otherwise, it isn't really.