Everyone on U.S. soil is subject to the law and rule of the courts to some extent, including diplomats. Diplomats are immune to prosecution for crimes, but that's different than being outside the jurisdiction of U.S. laws.

For example, in 2013 several Russian diplomats were indicted for Medicaid fraud: https://abcnews.com/US/russian-diplomats-scammed-medicaid-15.... They had diplomatic immunity, so the U.S. had to get the Russian government to waive the diplomatic immunity. But if Russia waived the immunity, the prosecution could proceed even though the diplomats had diplomatic immunity at the time the crime was committed.

Moreover, diplomats are subject to civil liability for commercial activities beyond their office: https://www.law.cornell.edu/wex/diplomatic_immunity. Diplomatic immunity doesn't protect them from suits in U.S. courts related to such activities.

So reading "subject to the jurisdiction thereof" to mean "subject to the law and the rule of the courts" proves too much. The U.S. and its courts have some level of jurisdiction over everyone and everything on U.S. soil. So that makes the phrase "born or naturalized in the United States, and subject to the jurisdiction thereof" redundant--the words "subject to the jurisdiction" aren't doing any work that the phrase "in the United States" isn't already doing.