I think part of the problem here is that we're getting hung up on illegal immigrants vs. legal immigrants, but from an economic standpoint, only the supply of unspecialized labor matters, not the source (aside from the burdensome regulations on "legit" workers compared to off-the-record workers). Also, since we all agree that dismantling the welfare state is the number one step to take in solving the "immigration problem," we can move past that point and pretend, for the sake of discussion, that it's "already done."
In that case, the question is, can a free market economy (without a welfare state) handle a surge of unspecialized labor? What are the consequences of that? I suspect that Dieseler is approaching this from a fundamentally protectionist point of view. I don't really want to get involved in the discussion other than to hopefully help reframe it constructively, but I wanted to point out that it's really just a special case of the question, "What, if anything, does protectionism achieve?" (I'm sure Ron Paul and the Austrians have some things to say about that.

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Granted, we live under different circumstances than the ideal: We do have a welfare state, we do have burdensome regulations that disadvantage "legit" workers, etc. However, in the case of the immigrants in Torchbearer's neck of the woods, the welfare state doesn't seem to be an issue according to him. Even in the case of schools, the moral burden does not fall on unrelated employers anyway, but on the government for extorting the money in the first place, and the illegals for taking advantage of schools others are being forced to pay for. When you narrow things down to Torchbearer's specific situation, aside from the regulations that "above-board" workers/employers must comply with, the issue really does just boil down to free market labor vs. labor protectionism. (Of course, this is ignoring the law too, but I think we're pretty used to having moral and economic discussions here regardless of what the current law says.)