Granted he was an evil dude. He was responsible for organizing the trains to pick up and take Jews to Auschwitz and other camps.
He claimed he was just a low level functionary. His highest rank was as a Lieutenant Colonel in the SS. He never participated in any killings although he may have witnessed one.
So if he had been tried at Munich, would he have received the death penalty? I’m thinking not. Many others who were far more involved in the holocaust didn’t receive the death penalty. Maybe life in Prison that would have been commuted.
Some examples:
https://en.m.wikipedia.org/wiki/Udo_von_Woyrsch (in charge of Einsatzgruppen in Poland)
https://en.m.wikipedia.org/wiki/Gottlob_Berger
https://en.m.wikipedia.org/wiki/Richard_Walther_Darré (developed racial policy)
https://en.m.wikipedia.org/wiki/Josias,_Hereditary_Prince_of_Waldeck_and_Pyrmont (Commandant of Buchenwald)
All of whom were Obergruppenfuhrer's, or Lieutenant General rank. There were of course many others. I think if he’d stuck around Germany, Eichmann would eventually have been brought to trial, but given a relatively light sentence.
SLD
Here are my general thoughts on the death penalty, first. I think the death penalty, at least in the US, is used way too frequently, but I am not 100% opposed to it in principle. I think there are extreme circumstances where a person is a threat to future society (i.e. individuals) whereby they ought to be executed. In order to fit into this category, to me, a person has to have demonstrated beyond a doubt, we have to be absolutely 100% sure they did some of these crimes against humans in the past and we have to be very certain there is immense RISK to humans in the future. To me, a person doesn't lose their rights NOT to be analyzed as a risk, except by violating the sacred fundamentals of society by murdering or similar crimes to others. After such, they lose certain rights and we can treat them like a risk, but we have to be 100% sure they did the things in question and then we can look at what could happen in the future. So, in Ted Bundy's case, I'd have been for his execution because he showed time and again he could escape and be a continued threat to society. In someone like Hitler's case, I would definitely think he was a threat to future society. Even if jailed, he could command others, or a resurgence of Nazism could one day break him out and the horrible acts could again occur. I do not normally ascribe to ideas of punishment, but instead rehabilitation.
To at least some extent, legal consequences such as the death penalty are also a deterrent to others to do such crimes. In the case of what we're talking about...a rogue nation committing genocide, I think consequences as a deterrent are also important. This means that some international human rights law violations in the extreme and future risk to others, but also that OTHER governments that could potentially go rogue have to observe what happens when attempting to commit genocide or other such horrific murders or similar against their own people. That is, a nation may make killing off the Jews or Muslims or Mexicans or whomever legal domestically. The world has to show all such nations that cannot be a thing to get away with--that there will still be consequences.
So, in the case of the Nazis, most major players of the Nazi regime ought to have been put to death because of the future risk to society, provided that their connection to the genocidal plans and actions were certain...as both a deterrent to their coming to power again to repeat such crimes and as a deterrent to others who might want to do the same. But the crimes of the Nazis were not limited to only the major top brass. Guards at the death camps and "cogs in the machine" like Eichmann were also involved in the murders, directly or indirectly. Any such low-ranking person who we could be sure of having been involved in mass murders directly and posing a future risk to the world also should have been executed. That means guards at the concentration camps.
I don't believe every single Nazi would need to be executed including those not directly shooting people in the head, but enough middle men and low-ranking persons ought to have been executed for two of the same reasons as above: to destroy the chances of Nazism coming back as the government of Germany and to act as a deterrent to other movements and countries considering similar crimes against humanity. Many hard core Nazi criminals could also alternatively be imprisoned for decades under the occupation and for some number that I cannot quantify, this could act as both a deterrent for future Nazi Germany and against other countries going rogue, and this is especially true if the leadership is extinct because those risks would be gone.
So this brings me to Eichmann now. If Eichmann were not a major player in the genocide or it could not be proven 100% that he was, then I would conclude that it is ambiguous. To me, as I described above, some sufficient number of Eichmann-like individuals ought to have been executed to act as a deterrent but once so many are executed, the benefit of executing individuals is decreased significantly. Some individuals who are not a future threat to society because there would be no resurgence of a Nazi regime need only act as a deterrent by imprisoning them. Likewise, some individuals according to their crimes, could have less consequences and be rehabilitated also showing the world that people CAN be rehabilitated in such situations.