Except,
1. Nobody suggests that DNA emerged fully formed - it was proceded by an age of RNA-based chemistry.
2. Pre-RNA molecules/structures would still have to be self-replicating, thus introducing a variety of selection pressures that would accelerate the rate of information retention for re-use in a subsequent iteration.
3. How much "data" is contained in a self-replicating compound? Is it more or less than in a sentence of Hamlet? Your Hamlet string has behind it a whole language, with idiom and abstract meaning, embeded in a complex cultural context. The compound only needs to specify how to make a copy of itself.