Jewish Zionist institutions effectively indoctrinate these Jews into joining the Israeli military.
This is exactly what I'm talking about. We all know that most Zionists are Jews, but it's not Judaism we're objecting to it's Zionism.
It's not the fact these institutions are Jewish that's making them indoctrinate Jews into joining the Israeli military, it's the fact these Jewish institutions are ZIONIST.
That's the whole problem.
Muddying the waters on this point does exactly two things: it signals anti-Semites that it's safe for them to be here, and it makes it easy for Zionists to point and say "see, it's Jews they hate"