The 5th Amendment also says ..."nor be deprived of life liberty or property without due process of law..." The Supreme has ruled that a judge cannot sentence a person to death unless the jury has recommended that they be put to death. You may remember that many people had their sentences reversed on this ruling. So how can a person be sentenced to death without a jury trial. Unknown. But the police can also kill you and say that you were a danger to them and they will never be brought to trial, see the story about the invalid being shot in bed by an officer who invaded his house through a second story window and is given a 30 day suspension which he says he will fight. So all in all we have lost our freedoms to the judiciary and the police.
Of course there are some who claim that due process of law has been followed in this case.
But, if the Schindlers never asked for a jury to hear the case, and were not informed by the judge or their attorney that they have that right, they may be able to use that as a basis for an appeal.