The mechanisms for producing new DNA sequences involves copying existing DNA sequences. Thus, the ones that exist are privileged over the ones that do not exist (yet), and the adjacent sequences are privileged over a random sequence.
> No DNA sequence is more likely than any other to exist.
It is far more likely for a DNA sequence very similar to my own to exist than a random sequence.
> we need to identify strong physical processes that produce useful DNA strands
We have already identified those processes! We know quite well how the machinery of DNA replication works.
If we care about the first DNA molecule to ever exist it's a very different question. We don't need to find a physical process that produces a modern DNA molecule from 'raw parts', rather one that takes not-quite-DNA and converts it into DNA.
> it's hard to imagine what those physical processes might be since the strands they must produce are extremely, unimaginably rare in practice.
Can you imagine slightly simpler DNA? Say just a bit shorter? What's the simplest molecule we might still call DNA, that is reproducing? Can we imagine machinery that would produce that?
I think it's very reasonable to think such machinery could exist, even if we don't know the exact mechanisms involved. We know that RNA can self-reproduce, and also produce proteins, so it's reasonable to think that machinery to produce RNA strands could evolve to produce DNA strands (for example).
The only involvement randomness has in this whole process are (relatively) rare and infrequent changes to self-replicating molecules, and (potentially) the initial formation of a self-replicating molecule.
It is irrelevant how many possible DNA sequences there are, or how much information is stored within them, as we know new sequences are derived from previous ones.