First of all, the trick relies on imputed 1NN/imputed kNN/imputed weighted kNN. And second of all, you still have to explain how imputed weighted kNN relates to the problem of taking a bunch of x/y pairs and predicting a novel y for some novel x. When you start to unpack all of that it does not look simpler to me than what I wrote.
For the weights, I'm not sure that's as mysterious as you make it seem. The most important thing they learn is, basically syntax. So you have 40k token types to choose from at each prediction, and the vast majority of them are statistically implausible because they are syntactically invalid. At some positions this restriction reduces entropy from ~15-16 bits (or 40k choices) to 2^8 (~200 choices) and in some places it reduces it "only" to 12 or 13 (~8k choices, common when the next token is "approximately a noun").
Once you winnow it down, it's much closer to a moderately big multiple choice problem. And much of that entropy lies just in the model's representation of word co-occurrences.