A big factor is not the willingness to do "dirty work", it is willingness to do the dirty work for substandard wages. US workers are loathe to do anything for minimum wage, much less for below minimum, which is where the immigrant workers fit in; if doing the "dirty work" were paid at a level of, say, 1-1/2 times minimum or higher, there might be an incentive for US workers to take those jobs...
All the talk of the immigrant workers has pretty much focused on the workers, themselves, and very, very little on the hiring entities, who are quite often violating Federal laws and regulations by hiring migrants. If there was a serious effort to hold the hiring parties responsible for adhering to long-existing laws, the incentive for immigrants to come to the US would be vastly reduced; people won't go where there are few or no jobs available to them. If you look at the way immigration laws regarding hiring are enforced, you would see a sort of 'blind eye' situation in play; there are laws prohibiting the employment of non-citizens or undocumented workers, but the enforcement against the employers is almost non-existent or very shoddy. As an example, there does exist a system for the verification of legal emplolyability status, but the use of the system, by law, is entirely voluntary: make it mandatory, make compliance a documented, and verified process, and, most importantly, make noncompliance punishable by hefty fines, sanctions, or, in extreme violation, jail time...
Why is the employment verification system voluntary and not mandatory? The employers who have profited from the use of non-eligible migrant labor such as farms, hotels, restaurants, and other places need workers for the "dirty work" have lobbied long and hard, at great expense, to keep their low-cost workforce and to avoid having to comply with existing laws...
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