Wouldn’t that imply that we’d find independent genesis of replicating molecules frequently even now?
However there may be slightly more requirements like the sequence has to be found in an oil film (bubble) up until it can form a skin/membrane, to protect itself, so even if it was happening on earth trillions of times every day no one would be there to discover it, and other life on the planet with a head start (bacteria, etc) would consume it before it made more 'progress' by evolution.
In the early planet there was nothing to "eat up" life as it was starting. Everything was on an equal footing.
For example, if I understand this correctly, a random jumble of DNA as you describe would have a 50/50 chance of twisting to the right or twisting to the left. And yet all life on earth uses DNA that is "right handed" twisting to the right.