You're assuming the existence of taxes. Perhaps with our current effed up system the whole no birthright citizenship/strong borders/deportation thing might be convenient. But in a free society with private property, you wouldn't need permission from any "government" to hire someone, pay them what you want, sell them a house, or otherwise voluntarily interact with them in any manner in which you mutually please, regardless of where they are from. There would be no "burden on the taxpayer" because there would be no taxpayers in the first place because taxation is theft, which wouldn't be tolerated in a voluntarist society.
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