The Vinogradov example seems curious. The "existential N" there is for encoding the intuition that the property holds eventually, e.g., that we don't know or do

The Vinogradov example seems curious. The "existential N" there is for encoding the intuition that the property holds eventually, e.g., that we don't know or don't care about which particular N along the number line for which P(n) is eventually true (forall n such that N < n). So in this interpretation, it is like using the quantified E like a Sorities/heap/vagueness argument, it is functioning metamathematically. Mentally I am picturing a number line, and then there's a vague area on the line where for everything to the right of the line it is colored "P". So why isn't this forall-exists usage a bit pathological, isn't it more that Vinogradov's theorem is "halfway" between either type of quantification? And that seems a very special case of how counterexamples break down rather than the general case? Could someone more mathematically knowledgeable explain this?