In order to do that the immigrants would have to pay enough in taxes to make up for the loss in utility by the preexisting residents. Some groups of immigrants, such as immigrants resembling the current Mexican illegals, may not be able to pay that. Furthermore, the same political groups who want more immigrants would oppose such a plan to the point where it won't be feasible.

And the fact that they have children in this country is part of what decreases utility for the people already in the country.