If replicators before DNA worked just as well with specific chirality as without it, wouldn't mutations have been just as likely to keep non-specific chirality just as it was?
I think you are just missing the picture of what is being described. There could have been mixtures of left and right handed DNA. Let's say theoretically that some left handed DNA managed to form a self-replicating uhmmm... "organism".
All it would have taken is for the right handed DNA based self-replicating "organisms" to survive and thrive, where the Left-handed precursors did not, to result in the reality we have now... all life happens to be right handed.
This is similar to the fact that in cosmology we are a matter universe. In reality matter and ant-matter are formed at the same time, so how can we have a "matter universe", technically it should have cancelled itself out. But matter production appears to have had a slight edge, when multiplied by the vast time and events we are talking about, it adds up to a rather large amount of matter.
Right handed DNA had the advantage for whatever reason, and its not like the existence of left handed DNA was going to screw with it... just not do anything at all.
As far as how did DNA strandish chiral forms get an edge up on forming, over straight chains or other bonding options, there has been a suggestion that chiral clays may have helped in this process. That is a naturally occurring phenomena which gives organic molecules that cling to them and then each other, a specific chiral template.
Haven't heard if this research has gone anywhere recently, but that was a plausible suggestion.