A few points:
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You have to accept the fact that there have been many criminals legitimately convicted by juries before the existence of DNA evidence. Even in an age of DNA evidence, one may be convicted of a crime through other evidence aside from DNA. -
The Defendant wants access to the evidence so he can pay to have it tested. You may think "What does the state care? It won't cost them anything." But if we allow this, then we must allow others to acquire evidence in their cases for testing, and if we allow that, then it's only fair that the state pick up the tab for those who can't afford to have it tested. This means that if there's a due process right to post-conviction DNA testing, we'll be picking up the tab for every jackass who wants his evidence subjected to more advanced testing, in the name of "making sure." The time to make sure is before the trial concludes. That's what our system is set up to ensure, as much as reasonably possible. -
Assuming you feel the same way I do, that there's little value in allowing every convicted criminal the right to have new DNA testing just because it happens to be available in their particular case, then maybe you can see the wisdom in allowing the states to craft their own post-conviction procedures to try and handle the rare instances of the falsely convicted so that the state isn't stuck having to revisit every criminal conviction made.