A "lower priority"? Shouldn't it be the opposite: the priority being that we don't tear up the families of our countrymen over paperwork issues?
I mention it because some people think "Oh, we're throwing the parents of American citizens out" --no it's the parents of someone who both hold foreign as well as American citizenship. So one could as equally say they're deporting the parents of a Serbian or Brazilian citizen, but it'd be most accurate to say the parents of a dual Brazilian and American or dual Serbian and American citizen.
So your parents were not more American than say they were Irish [if their parents were still legally Irish at their births]. They were the same. But we're trying to say they are more American than they are Irish and by law they would not be.
And replace it with what, exactly? That clause [0] is, after all, what created uniform national citizenship, and eliminated varying and discriminatory state citizenship policies. Even if one agrees that the "anchor baby" problem is something that warrants limiting citizenship to combat (highly contentious proposition itself), simply blowing up the entire post-Civil War model of US citizenship is a rather brute approach with wide ranging consequences outside the immediate target area.
[0] "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."