This is a reasonable conjecture. I think I have seen clarifications from Senator Howard that indicated he didn't care if they were naturalized or not, which clarifies this against the meaning offered.
However, Chief author of the 14th amendment Representative John Bingham makes it very clear that he has no intention of granting citizenship to the children of foreign nationals.
Who are natural-born citizens but those born within the Republic? Those born within the Republic, whether black or white, are citizens by birth natural born citizens. There is no such word as white in your constitution. Citizenship, therefore does not depend upon complexion any more than it depends upon the rights of election or of office. All from other lands, who by the terms of your laws and compliance with their provisions become naturalized, and are adopted citizens of the United States; all other persons born within the Republic, of parents owing allegiance to no other sovereignty, are natural born citizens.
And here is another quote from John Bingham:
..[I] find no fault with the introductory clause, which is simply declaratory of what is written in the Constitution, that every human being born within the jurisdiction of the United States of parents not owing allegiance to any foreign sovereignty is, in the language of your Constitution itself, a natural born citizen. . . .
It is reasonable to believe that this is the opinion he shared with the other members of congress when the Amendment was being drafted and debated.
Now that quote is a much stronger argument for the proposition that the children of illegals should not get automatic birth-right citizenship. Personally, I think Congress could resolve the question with a simple statute, stating that foreign citizens are not considered "subject to the jurisdiction of the United States" for the purposes of the 14th amendment unless they are legal permanent residents. And then include a clause in the statute denying the judicial branch jurisdiction over that statute.