I know about the role of the Vatican. Why does one not understand why the Israelis did what they did? The Mossad went after those who went underground and took them out in several ways to include kidnapping (that term could be construed as harsh) or just outright killing them where they were found.
Israel chased nobody. The Mossad investigated no one. Individual Jews found these men of whom you speak and brought them to justice.
Eichmann is seen by an old blind Jew who sees more through the eyes of his daughter than men with two good eyes. He pressures Tel Aviv until Mossad is forced to send an abduction team and return him to Israel for public trial in one of the most anti-climatic events in history. The problem with the Eichmann operation is it created true paranoia in the other Big Names and they dove deep underground after that rather than living as an open secret in Juan Peron's Argentina.
Serge and Beate Klarsfeld spend their life hunting Jews. They find Herbert Cukurs of Rudolph Lange's Riga Einsatzgruppen in Uraguay running (not surprisingly) a flying boat service and pinpoint his location for a Mossad hit squad to apply Special Treatment. They find and spotlight Klaus Barbie in Bolivia and his friends in CIA did nothing to stop his extradition when kidnapping proved non-viable. Amongst others.
Simon Wiesenthal, the most famous Nazi hunter of them all, merely collects data and does very little with it other than share data that leads to a few a few high profile cases involving Eichmann, Fritz Stangl, and that sadistic woman from Madjdanek Camp. He knows where Mengele is hiding, he spends a good deal of time documenting Bormann's hidden personas in South America, but knowing something and being able to do something about it are two different items.
Do you think Augusto Pinochet, whose unofficial director of security is Walter Rauff, Heydrich's personal aide, is going to give up the Nazis hiding in the fortified 'cult' compound of Colonia Dignidad? Or Juan or Isabel Peron? Alfredo Stroessner is going to give up Gestapo Muller, his head of security, or Dr. Mengele who is the exiled Nazis doctor of choice? No. And an individual who goes seeking Nazis in these places takes his life into his own hands and risks Night and Fog for his efforts or imprisonment until the next time the Caravan of Death comes to town.
You have a handful of individuals who see justice, the vast majority walk free and die in their own beds.
However. 'Vengeance is Mine, sayeth the Lord.' I have no doubt these men met a much worse fate than any humankind can devise for what was done to His Chosen People.