Blaming the British White Paper of 1939 for preventing Jews escaping the Holocaust is less than accurate. The restrictive immigration policy went into effect within a few months of Germany invading Poland... a little late to be thinking about going to Palestine.
Prior to the White Paper of 1939, my understanding is that all a Jew had to do to get a visa to Palestine was purchase land. I don't recall there being a minimum purchase size either. Why didn't more European Jews see the writing on the wall prior to 1939 and at least buy a plot of land in Palestine as an insurance policy?
Furthermore, other nations had unfilled visas. The Dominican Republic in particular had 100K visa's available for European Jews, almost none of which were filled. That the Dominican Republic had so many unfilled visa's should bear witness to the state of denial that European Jews were in.
I think at the beginning of the war only a very few people thought that a civilized European country could embark on something as evil and barbarous as the Holocaust.
You probably need to brush up on your history a bit. Eichmann was allowing Jews to buy their way toward Palestine from Vienna as late as the fall of 1940. And Jews in the pipeline were still trying to run the gauntlet into 1942.
After the war started, the Soviets probably killed the most Jews trying to reach Palestine, the Germans were second, and the militant Jews were third in the number killed. But British hands certainly were not clean.