So back in the days when a person born to U.S. Citizens in a rural area never had his or her birth formally recorded, they weren't subject to the jurisdiction of the U.S.?
“They” = “he or she”
Or put it this way, if I drive on I-10 from Louisiana across the Texas state line, but Texas has no record of my presence, does that mean I’m not subject to the jurisdiction of the State of Texas?