The story of the Zimmermann telegram is indeed an interesting tale.
My grandfather was in the Secret Service back in those days. He was sent to Mexico as part of an investigation as to what exactly the Germans were up to.
One of MY best friends (50 years ago) as a kid was named Zimmerman...nope, they were CATHOLIC.
Zimmerman is a German name (ending in 'mann'). There were a number of Jews in Germany, some named Zimmerman. Most Zimmermans probably were not Jews. Often the N at the end was dropped when they came to America.