Would you be willing to consider, i.e. not discard, a decoder that used the human DNA as merely a list of indexes, downloading the required genes from some sort of internal lookup table?

By changing the lookup table, one can dramatically change the resulting organism; and having a different result for every viable human DNA is merely a resut of having a large enough lookup table. It would be, to extend your metaphor, like a text-translation algorithm that returned the Declaration of Independance if given as input Alice in Wonderland, and returned Alice in Wonderland if given Hamlet.