I am not sure a wider opening of Palestine would have saved that many more Jews.
The Jewish population of the Western European countries had literally swollen during the interwar period, taking refugees from Central and Eastern Europe. Most of them were simply aspiring to integrate themselves into those societies when the war caught up with them. It is true that some quota's had been put on Jewish immigration after the influx from Germany grew so much following the Kristallennacht, but they still numbered in the tens of thousands. It was also the moment when the Nazi started ransoming the candidates for departure, making it even more difficult for them to depart at all, whatever the destination.
The US remained fairly open to Jewish immigration throughout the period, and did indeed attract lots of refugees. Any Jew who wanted to leave a dangerous country would stand a reasonable chance of getting into the US. Many preferred to stay in Western Europe, as they wanted to stay close to their country of origin or had family connections there.
I would be of the opinion that Jews who wanted to leave but did not do so, didn't do it for lack of an opportunity to go to Palestine (I'm talking before the war, here - the attitude changed radically afterwards), and that those who settled in to-be-occupied territories did not make this choice either for that reason. It may be that a number of committed Zionists were indeed prevented from joining Palestine and were hence caught by the horror of the Holocaust, but you'd probably be talking in the tens of thousands, maybe a couple of hundred thousands at the most... And I'm not even talking about simply the capacity of this small land to absorb such an inflow of allochtone population over such a short time span.
As an answer to the previous question, Zionists were mostly the secular entrepreneurial type. Many religious Jews were anti-Zionists at the time.