Strict constructionists would listen to the intent as well. We've been over this... You cannot say that because you read it that way that is what they will decide, if there is written documentation that it was not the intent of the Amendment to allow illegal immigrants to sneak in and birth citizens, written from the time period, then they will likely side with the intent, especially if it is due to evolved language.
In this case you get the framers, particularly Senator Lyman Trumbull, who emphasized that "jurisdiction" meant owing allegiance to the U.S. For example, children of foreign nationals temporarily in the U.S. might not be considered under full U.S. jurisdiction if their parents owed allegiance to another country. During the 1866 debates on the 14th Amendment, framers like Senator Jacob Howard clarified that "jurisdiction" meant "full and complete jurisdiction," excluding those who owed allegiance to foreign powers or were part of semi-sovereign entities (e.g., tribes).
The amendment was partly a response to the Dred Scott decision (1857), which denied citizenship to African Americans. The Citizenship Clause aimed to ensure that freed slaves and their descendants, born in the U.S. and subject to its laws, were citizens, but it wasn’t intended to automatically include everyone physically present.
If they listen to these things as to inform on intent of the time period due to evolving language they may very well decide differently than you think they will if they are strict constructionists.