I've heard dozens of Germans pronounce it with a soft “s.”
I've never heard it pronounced “tset” with an inflected “t.”
My own German is very basic, so I can't dispute what you say.
But I am surprised to hear all this.
By the way, at least one native born German thought that “zi” comes from Sozialist, which is where I got the idea.
You're correct. From Wikapedia...
The National Socialist German Workers' Party (German: Nationalsozialistische Deutsche Arbeiterpartei, abbreviated NSDAP), commonly known in English as the Nazi Party (/ˈnɑːtsi/), was a political party in Germany active between 1920 and 1945. Its predecessor, the German Workers' Party (DAP), existed from 1919 to 1920. The term Nazi is German and stems from Nationalsozialist, due to the pronunciation of Latin -tion- as -tsion- in German (rather than -shon- as it is in English), with German z being pronounced as 'ts'.
The disparate handling of the "ts" sound may trace to where your family and my German teacher were from. Herr Scheck was from Stuttgart, so I learned German with a Bavarian dialect.