Indeed, it would. It also would cause a political revolt. Such a revolt doesn't occur at the lower end of the income scale because the natural representative of the interests of native low-skilled workers -- the Democratic Party -- has bought into high levels of immigration in the hopes of getting new voters.
Bush's failings at the border mirror his failings in Iraq. In both places, he underestimated the need for security and order and has undertaken a push for them only belatedly. In both places, he was motivated by a good-hearted belief in the essential fungibility of people. He thought that Iraqis naturally would have the same desires as Westerners; and on the border, he assumes that Mexicans are seamlessly interchangeable with Americans, since they seek employment here.
In Iraq, he has tried to compensate for his mistakes. On the border, he seems hopeless.