You keep touching on the crux of the matter. As designed, our criminal justice system was supposed to accord the benefit of the doubt, a reasonable doubt, to the accused. Better that ten guilty men go free than one innocent man hang.
Unfortunately, the problem is not with a truly judiciously administered punishment for capital crimes upon conviction, it is found in the rush to 'solve' s crime, one which lets the real perpetrator go free while the press, the clamoring mob (the public), and the politicians and judges are sated in their quests for feelings of safety and vengeance and closure. It is that rush to judgement which corrupts the system, that clamoring to feel 'safe' when the result might be anything but safety.
We need to return to those Constitutional protections for the accused, good police work, honest courts, and rewards based on getting it right rather than just 'doing something'.
Then, without the politics in the middle of the judicial process, operating as originally intended, the judicious use of the death penalty would in some cases be warranted.