The logic is simple, but results in finding that Wong Kim ark was wrongly decided. Just a quick summary of WKA - the majority stretched the phrase "subject to the jurisdiction" to include aliens temporarily on US soil, except ambassadors and hostile armies.
The logic of applying the preamble is that children of aliens aren't part of "we the people," because they aren't part of "we the people's" children (posterity). The document secure to "we the people" and our posterity, not to the posterity of aliens. Therefore, the posterity of aliens aren't citizens, and if they aren't citizens, they can;t be natural born citizens.
Interesting take! However, doesn’t that leave all foreigners/non-citizens on US soil exempt from the Constitution? Not just part of it - but all of it? After all, your interpretation of the Preamble would mean the Constitution only applies to citizens and their descendants.