I think the problem is that, despite a prisoner's constitutional right to habeus corpus, once you are convicted you no longer have the presumption of innocence.
Add to the fact that the convicted was convicted by a jury of their peers. Therefore, to overturn the conviction, the courts not only have to recognized and acknowledge their own error, they need to overturn the jury's decision. And the system places such a high value on jury decisions that it doesn't treat overturning them lightly.
This means that there is a huge uphill battle to climb. The presumption is that you are guilty, because you were convicted beyond a shadow of a reasonable doubt. You are no longer in a position of defending against a prosecutor's case where they need to prove you guilty and you have no burden of proof what-so-ever. You are now in a position where you need to prove your innocence.
I'm not sure what the process is like when there is exculpatory evidence. I would hope that as long as the evidence has the slightest hint of merit that the case can be reopened. In the case of the article, however, (which I will readily admit that I didn't study in its entirety since it was pretty TL;DR, so I skimmed and skipped parts) it doesn't sound like there is new exculpatory evidence, it sounds like the DA is unconvinced by the evidence and that if the trial were to occur today, that evidence would have been given much stricter scrutiny than it was at the time. It sounds like we need better ways to rexamine cases when standards of evidence change. To offer a made-up hypothetical, imagine that polygraph tests were admissible at one point and then it was deemed they are unreliable and are no longer admissible, could people who were convicted on polygraph evidence demand an appeal on that ground? I'm not a lawyer so I don't know, but I would hope so.