Of course it isn't actually integral, or there wouldn't be Jewish people protesting israel. Lots of people have one without the other.

i'm not very up on my differences between the torah and the christianized old testament, but i'm pretty sure there's some "god said kill all these people" in there.

of course I haven't heard of anyone saying palestinians are actually amalekites, and we've all seen the video of the israeli cops shoving an old hasidic man protesting the genocide

Strictly, Zionism is opposed to religious Judaism, which is very clear on the Jews not being allowed to have a Jewish state in Eretz Yisrael until the Messiah comes (it's why most orthodox Jews are against it by default).