The short answer is politics. Here’s a Wikipedia article on the History of US Immigration.
Over time, the laws regarding immigration have varied rather considerably, based on on the perceived need for immigrant labor, the national mood toward outsiders, and often stereotypes attached to people of varying national origins.
If you visit the Ellis Island museum, something I highly recommend if you are in New York, the main exhibit is on the immigration requirements during the primary time Ellis Island was active. Although I don’t know exactly when Corleone was supposed to have immigrated, what you describe sounds like the Ellis Island process. However, once you got through Ellis Island, you were allowed to come into the US, not become a citizen.
Here’s a blurb from the Ellis Island website on what an immigrant would go through:
If the immigrant’s papers were in order and they were in reasonably good health, the Ellis Island inspection process would last approximately three to five hours. The inspections took place in the Registry Room (or Great Hall), where doctors would briefly scan every immigrant for obvious physical ailments. Doctors at Ellis Island soon became very adept at conducting these “six second physicals.” By 1916, it was said that a doctor could identify numerous medical conditions (ranging from anemia to goiters to varicose veins) just by glancing at an immigrant. The ship’s manifest log (that had been filled out back at the port of embarkation) contained the immigrant’s name and his/her answers to twenty-nine questions. This document was used by the legal inspectors at Ellis Island to cross examine the immigrant during the legal (or primary) inspection. At the museum there are also examples of cards used to prove that immigrants were literate in at least one language (including some highly obscure ones), which I believe was a requirement that came under a change of law during the period of Ellis Island’s operation.
In the 1920s the immigration laws changed to make immigration much more restrictive. Subsequent changes of law added to the burdens of those trying to immigrate.
Today, immigration is such a difficult political subject that there is little legislative will to improve or change what almost everyone agrees is a seriously broken system. The administrative system for processing immigrants is well known as a Kafka-esque disaster, but because those using it are, for the most part, well, immigrants with little political power, there is little incentive to improve it. Those who have the economic resources will hire lawyers and advisors who can navigate through the bureacracy (and even then, it sucks), while the little guys get chewed up.