Not a bad idea, though.
No it wouldn't. The 14th states:
"All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside."
This does not apply to foreign nationals, whether in this country legally or illegally. As Jacob Howard, coauthor of the citizenship clause of the 14th amendment stated in 1866:
"Every Person born within the limits of the United States, and subject to their jurisdiction, is by virtue of natural law and national law a citizen of the United States. This will not, of course, include persons born in the United States who are foreigners, aliens, who belong to the families of ambassadors or foreign ministers accredited to the Government of the United States, but will include every other class of persons."
The citizenship clause was thus clearly meant to prevent emancipated slaves from being denied citizenship, not to provide an end run around our immigration policies.
On a recent FR discussion I read that the anchor baby status thing arose from law to keep southern states from disenfranchising recently freed slaves as voters