I think with DNA evidence, we have gotten away from executing innocent people. However, nothing is infallible. Without DNA, we shouldn’t seek the death penalty. Eye witness testimony is unreliable.
Most appeals concern representation issues and mental capacity; things that could convert the death penalty sentence to life, not necessarily for overturning the guilty verdict. Two years should be plenty.
Lastly, I don’t believe that the appeals process is about assuring that justice has been served, as it is about lawyer’s games and egos.