For DNA, at least, it could have come from RNA (as per the link in my last post).
While I don't think the pre-biotic problem is solved at all, we have a lot more models of how it could have happened than you seem to credit - this is after all a huge research area.
For example, here is one [0], and here is a whole journal issue on the subject [1].
I found these by searching for 'evolution of DNA' and 'evolution of RNA'.
Now, these models all include some randomness, but in no way does anyone assume "unique random initial states are continually occurring... letting the law of large numbers eventually "find" winners"
The models show plausible environments where pre-biotic synthesis of RNA (or RNA pre-cursors) can occur, and stabilise.
This model you keep bringing up - randomly selecting a molecule from all possible combinations of atoms and saying 'enough time will get you one that works' - is not mentioned anywhere that I have seen. Perhaps some lay-people (of which I am definitely one!) believe it, but as you point out it is so obviously implausible it falls down on first inspection.
There are other models (lots of them!) and they don't rely on this pure randomness.
While I don't think the pre-biotic problem is solved at all, we have a lot more models of how it could have happened than you seem to credit - this is after all a huge research area.
For example, here is one [0], and here is a whole journal issue on the subject [1].
I found these by searching for 'evolution of DNA' and 'evolution of RNA'.
Now, these models all include some randomness, but in no way does anyone assume "unique random initial states are continually occurring... letting the law of large numbers eventually "find" winners"
The models show plausible environments where pre-biotic synthesis of RNA (or RNA pre-cursors) can occur, and stabilise.
This model you keep bringing up - randomly selecting a molecule from all possible combinations of atoms and saying 'enough time will get you one that works' - is not mentioned anywhere that I have seen. Perhaps some lay-people (of which I am definitely one!) believe it, but as you point out it is so obviously implausible it falls down on first inspection.
There are other models (lots of them!) and they don't rely on this pure randomness.
[0] https://phys.org/news/2018-01-chemical-evolution-dna-rna-ear...
[1] https://www.mdpi.com/journal/life/special_issues/evolution-R...