It seems to me that your question is asking why Allied nations chose to send their Jews to create Israel, rather than integrate them post-WWII. The problem with that question is that it rests on a faulty assumption: namely that Israel was created by refugees from the Holocaust. That would be an incorrect assumption.

The creation of Israel was completed primarily by immigrants who had arrived in the area over the previous 60 years. The First Aliyah, dated to around the 1880s, was not the first movement of Zionist Jews seeking to establish a Jewish community and later state in what became Israel. However, it was one of the first organized movements that resulted in a serious migration of people (roughly 25,000), and was followed by multiple further periods of immigration also known as "Aliyahs". The Second Aliyah, usually dated as starting in 1904 and running through 1914, brought many of the most recognizable figures in Israel's founding canon to the area, such as David Ben-Gurion (widely regarded as Israel's "George Washington", and its first Prime Minister). Each of these waves of migration were largely motivated by rising antisemitism in Eastern Europe and Russia, resulting in infamous pogroms where Jews were massacred (the most commonly recognized being the Kishinev pogrom, where 49 people were slaughtered on Passover). The rise of antisemitism pushed many Jews into Western Europe, but a rising tide of Jews believed that salvation lay in establishing a Jewish state, resulting in the migrations I'm describing.

It's true that the rise of Nazism brought many Jews to what became Israel, but the bump came before the Holocaust, not after. It was not a decision by the Allied states. The rise of the Nazi party in Germany led to the emigration of over 35,000 Jews immediately from Germany (which was home to around 520,000 Jews in 1933). By September 1939, the start of WWII, over 400,000 Jews had left Germany and German-annexed Austria. Around 60,000 of them went to what later became Israel, while the rest fled to other parts of Europe or the United States primarily.

There were already well over 150,000 Jews in the area by 1931, however. The Jewish community had already set up many governing-style institutions, as well as a paramilitary force dedicated to defending Jewish communities (and one much smaller paramilitary force that was dedicated to offensive operations against British and Arab forces and which sometimes targeted civilians). By 1947, there were still hundreds of thousands of Jews living in displaced persons camps, many of them actually desiring to go to Israel to start their lives anew, and many having survived the Holocaust and having no desire to continue living in Europe. However, the total number in these camps was around 250,000, and in 1947 the Jewish population in what became Israel was already over 600,000. The movement to create Israel had a magnetism of its own, well beyond any Allied desire to empty the displaced persons into Israel to ease the burden on European states and to meet the desires of many of these displaced persons. It was therefore fairly logical for the Allied states to say that any Jew who wanted to enter Israel following the Holocaust should be able to do so. As the UN Special Committee on Palestine (UNSCOP) put it in its report recommending partition in 1947:

The distressed Jews of Europe, together with the displaced persons generally, are a legacy of the Second World War. They are a recognized international responsibility. Owing however to the insistent demands that the distressed Jews be admitted freely and immediately into Palestine, and to the intense urge which exists among these people themselves to the same end, they constitute a vital and difficult factor in the solution.

The solution of the problem of Palestine is rendered more difficult by the fact that large numbers of Jews, and especially the displaced and homeless Jews of Europe, insistently demand the right to settle in Palestine, on the basis of the historical association of the Jewish people with that country, and are strongly supported in this demand by all of the Jews encountered by the Committee in Palestine.

Thus UNSCOP, and the Allies, recognized that many of the Jews in these refugee camps were quite willing to move to what became Israel, and sought to do so urgently. That was a big part of why they wanted to open up that possibility, and why this type of immigration proposal was urged as part of a two-state solution in the UN's proposed partition plan of 1947. However, that plan was never implemented, and the establishment of Israel took a much different turn when, following the 1947 civil war between Jews and Arabs, Israel declared independence and was invaded by the Arab states. During the period between 1945 to 1947, many Jews fleeing these displaced persons camps were already being ferried into what became Israel. One famous example was the Exodus Affair, where UNSCOP members were taken by Zionist officials to ports to watch as a ship (the Exodus) carrying Holocaust survivors was set to arrive in Haifa. It was meant to arrive despite a blockade of the area put in place by the British, who ran the area at the time. The British boarded the ship, steered it into Haifa while being watched by UNSCOP officials, and sent the ship back to France (where it had originated). This was an international debacle; the French government said that it would only unload the passengers if they wanted to be unloaded, and they refused to go back to France. The French government refused the British demand to unload the Holocaust survivors into France, so the British decided to sail the ship to Germany, resulting in the occupants launching a hunger strike and refusing to unload. 100 military police and 200 soldiers were sent to force the Jews to disembark into British-controlled Germany, and the whole thing got a whole lot of press attention. The British had barred Jewish immigration to the area to meet Arab demands, since Arabs had previously rioted and revolted over the rise of Jewish immigration to the area, and many Jews ended up being placed in prison camps in Cyprus when they were caught illegally trying to enter the area where Israel ended up.

So why not just grant asylum or citizenship? Jews didn't want it. They wanted to go to what became Israel, their historic homeland, and take that route forward. The debacles that resulted from trying to prevent that made the British government look increasingly bad. That, coupled with pressure from the international community for a resolution to the dispute there, resulted in UNSCOP's search for a proposal that allowed for the creation of a Jewish state where the Jewish refugees could go. The logical solution, again, was for that state to be where 600,000 Jews were already living and creating institutions of governance, and in the area they regarded with historic and religious significance. It was not until after Israel was created that many of the displaced persons from the Allied states actually got to Israel, alongside the hundreds of thousands of other Jewish refugees fleeing Arab persecution in states as far-flung as Iraq and Egypt. It is therefore unsurprising that Israel's Jewish population nearly doubled within 3 years of its establishment, as Holocaust survivors and Jews from Arab states arrived in Israel.