According to wiki:

The Bracero Program (1942-1964) was a temporary-worker importation agreement between the United States and Mexico. Initially created as an emergency procedure to alleviate wartime labor shortages, the 1942 program actually lasted until 1964, bringing approximately 4.5 million legal Mexican workers into the United States during its lifespan.

The Bracero Program expanded during the early 1950s, admitting more than 400,000 Mexican workers for temporary employment per year until 1959 when numbers began a steady decline. While illegal immigration was a concern of both the United States and Mexico, the Bracero Program was seen as a partial solution to the upsurge of undocumented worker entries.

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This takes care of the argument that we need illegal immigrants to do the jobs American citizens won't do. Is there any reason why this can't be one of the tools to solve the illegal immigration problem?