The legal theory is actually sound -- the convictions were overturned based on the prosecutor's belief that if this evidence had been available at the time, the outcome of the trial would probably have been different.
This is not to defend the D.A. here -- in fact, he had absolutely no reason to throw out the other convictions that were handed down in that trial (for assault and battery by these young thugs during their "wilding" spree that night).
Interestingly, the defense attorneys truly botched the case because they had a perfect alibi for their clients. If they had constructed the timeline properly for that night, they could have shown that the thugs could not have raped that woman because they were busy assaulting people elsewhere in Central Park at the time.
In other words, they exposed the defendants to sentences of 10 years or more because they didn't want to effectively plead guilty to lesser offenses.