Reason for KeyNotFoundException in Dictionary initialization

Your first piece of code is using a collection initializer, which doesn't use logical assignment, but instead is intended to call Add on an existing collection. In other words, this:

var x = new Dictionary<string, List<int>> {
    ["a"] = {1},
};

is equivalent to:

var tmp = new Dictionary<string, List<int>>();
var list = tmp["a"];
list.Add(1);
var x = tmp;

Hopefully it's obvious from that why the second line of the expansion would throw an exception.

Part of your error in reasoning is:

albeit that {1} is a perfectly well-formed array

No, it's not. The syntax {1} means different things in different contexts. In this case, it's a collection initializer. In the statement:

int[] a = { 1, 2, 3, 4 };

it's an array initializer. That syntax only creates a new array in an array declaration, or as part of an array creation expression, e.g. new[] { 1, 2, 3, 4 }.