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Short answer
The Palestinian refugee problem is based on the fact that many Arabs voluntarily fled Israel in 1948, and were not welcomed in any Arab countries. Israel did not cause the problem, and UNRWA, like any bureaucracy, has done its utmost to keep its cause of existence viable. Most Arab countries do not want a direct relationship with the Palestinians themselves but try to support their cause to negate the State of Israel.
Long answer
Most of the Arabs living in Israel left immediately after it declared independence in 1948 and during the subsequent war started by the Arabs and neighboring Arab countries. They have not been accepted as immigrants by any other countries, including those that started the war against Israel, and thus remained classified as refugees. Let’s also talk about UNRWA, the only UN agency dedicated to dealing with a particular group of refugees: Palestinians. When Arab states refused to take in Palestinian refugees after the 1948 War of Independence, UNRWA expected that these refugees would return to their homes within 2 years. Yet with Arab states’ encouragement, UNRWA ended up not assisting refugees’ assimilation into their new countries and worsened the situation. Nowadays, most Arab countries do not want a direct relationship with the Palestinians themselves.