In this case it is hardly an analogy. With respect to Information Theory, compression and machine learning are essentially the same thing [0]. If you can understand why taking a hi-res photograph of a lo-res image does not create more information (and therefore cannot be used to make a hi-resolution poster) then it's essentially the exact same logic explaining why outputs of a model cannot be used to improve that same model.
More to this point, which other comments have already pointed out, what's interesting about this post is it asks whether or not very, very large data sets can be reasonably compressed by models such that they can simulate training another model while requiring less data.
From an information theoretic perspective this latter case is both completely sound and of potentially big practical importance.
I do have to say, I'm a little surprised that basic information theory is no longer common knowledge among the HN community (more-so surprised at a resistance to it), especially among people interested in Machine Learning. Out of curiosity, do you have a background in ML and/or Comp Sci?
0. https://en.wikipedia.org/wiki/Data_compression#Machine_learn...