Er, no. The Slaughterhouse Cases address children of LEGAL aliens. Allow me to excerpt the critical sentence.
Until they take an oath of US citizenship, in which they renounce citizenship of a foreign State, resident aliens, even legal immigrants, are subjects of a foreign State. The children of legal residents were not to be citizens. Should the parents be naturalized, or the children reach the age of majority and then be naturalized, the Congress would be charged with the determination of the means. The change to the familiar latter day understanding of the phrase is a construct of the Roosevelt Court, starting with Bridges v. Wixon, 326 US 35 (1945). There was a thread on this topic some months ago you might wish to read.