Agreed. Still, even if the DNA was on record and then Kirk could locate where it was on record and then have it transmitted to the Enterprise, we already know what would have happened.
Spock, "I think we have a match"
Kirk, "But not an exact match"
Basically, the story was going to make up any excuse why Kodos could not be identified with 100 percent certainty. So the easiest thing to assume is that the DNA evidence was not available and there were no other means to identify him, other than the voice recognition, which was not able to work perfectly because of the long time period and "the actor's" ability to change his voice somewhat. Yes, Kirk lied when he said, "disguising your voice will make no difference" because it seems to have made at least a small difference which was enough for him to reject the evidence.
Forgetting that 1965 was not a time when writers would be thinking about DNA evidence and viewers would not either, we can still make good excuses for why not all DNA is available. We could imagine that there is a privacy issue and long ago laws were enacted to protect DNA and people were reluctant to submit samples. Or, given the size of the inhabited part of the galaxy, maybe remote outposts do not go through the trouble to record DNA systematically. Perhaps it is done regularly on Earth in major cities, but remote areas of Earth (if they even exist) and backwater planets? maybe no. Maybe the records are there, but for some reason are not organized and centralized making it hard for a Starship to get the information quickly. Maybe within a few weeks of Kodos dying they do in fact track down a sample or a record and positively identify him and the history books can now be corrected.
We can imagine that in another 55 years from now, new identification methods will be known that we are not even thinking about now. People then will also debate about that technology in relation to this episode because TOS will still be watched and talked about, which I think is amazing.