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No, it's classified as the "new species" pokémon because, at the time of the RBY generation, it was a recently discovered pokémon, known previously only from legends (not to mention being a secret unlocked after collecting every other pokémon in those games). And what the dexes say is that Mew might actually be the ancestor of all pokémon, none of them imply what you are saying, instead only mentioning that it has all the DNA, not that it uses it.

However, with the arrival of Arceus, which challenged Mew's position as the original pokémon, I actually came up with an alternative theory for Mew's role in the pokémon mythos: he's the legendary ancestor of all non-legendary pokémon. After all, pokémon scientists wouldn't have much DNA from legendary pokémon to compare Mew's with, would they? Perhaps other legendaries' DNA is not present in Mew, and they just don't know it.