The Zionist movement was a secular movement founded at the end of the 19th Century, based on the view that anti-semitism made a Jewish homeland necessary. Given the history of Nazi Germany it would be hard to say that they were wrong.
The Zionists were offered - and considered - land in Uganda offered by the British Government. At this stage Zionism was a peaceful political movement encouraging Jews to move to and settle in Palestine, at that time ruled by the Ottoman Empire. The hope was to get Ottoman support for a Jewish homeland in Palestine.
From there things got slowly worse. The Jews moving in, firstly from Eastern Europe (especially after the pogroms in Russia) and later from other countries had their own prejudices. As did the Palestinian Arabs. The Zionists wanted political power and the Arabs objected to any concessions. Both sides did achieve some concessions (including a restriction on Jewish immigration, in the years leading up to and shortly after WWII) The first reference to serious violence I have found is Arab attacks on Jews, over Jewish use of the Western ("Wailing") Wall in 1929. Jewish attacks on Arabs seem to have mainly started in 1936 when the Haganah organisation moved from defending Jews to preemptive attacks. Around that time a related organisation, Irgun turned to terrorism and began attacks on British and Arab targets.
In 1948 the UN partitioned Palestine between Israel and Jordan (then Transjordan). The armies of Jordan, Syria, Egypt, Lebanon and Iraq attacked Israel. Despite being outnumbered and relatively poorly equipped isreal defeaed the attack and seized more territory (taking the opportunity to expel many Palestinian Arabs.
So yes, the Zionists did want a chunk of the country. But mostly they wanted it by peaceful political means. And many were secular, not religious.
And you re right that the Jews were never completely banned from immigrating - but isn't that what you are suggesting SHOULD have been done ?