I do not care what a Senator said. The Constitution states otherwise.
You might want to read what the writers of the 14th Amendment had to say on the subject.
Before the adoption of the Fourteenth Amendment, citizens of the states were automatically considered citizens of the United States. In 1857, the
Dred Scott v. Sanford decision had held that no black of African descent (even a freed black) could be a citizen of the United States. The Fourteenth Amendment was thus necessary to overturn
Dred Scott and to settle the question of the citizenship of the newly freed slaves. The Fourteenth Amendment made United States citizenship primary and state citizenship derivative. The primacy of federal citizenship made it impossible for states to prevent former slaves from becoming United States citizens by withholding state citizenship. States could no longer prevent any black from United States citizenship or from state citizenship either.
The Civil Rights Act of 1866 had previously asserted that "All persons born in the United States and not subject to any foreign power, excluding Indians not taxed, are hereby declared to be citizens of the United States." The immediate impetus for the Fourteenth Amendment was to constitutionalize and validate the Civil Rights Act because some had questioned whether the Thirteenth Amendment was a sufficient basis for its constitutionality. A constitutional amendment would also have the advantage of preventing a later unfriendly Congress from repealing it.
One conspicuous departure from the language of the Civil Rights Act was the elimination of the phrase "Indians not taxed." Senator Jacob Howard of Ohio, the author of the Citizenship Clause, defended the new language against the charge that it would make Indians citizens of the United States. Howard assured skeptics that "Indians born within the limits of the United States, and who maintain their tribal relations, are not, in the sense of this amendment, born subject to the jurisdiction of the United States." Senator Lyman Trumbull, Chairman of the Senate Judiciary Committee, supported Howard, contending that "subject to the jurisdiction thereof" meant "not owing allegiance to anybody else...subject to the complete jurisdiction of the United States." Indians, he concluded, were not "subject to the jurisdiction" of the United States because they owed allegiance—even if only partial allegiance—to their tribes. Thus, two requirements were set for United States citizenship: born or naturalized in the United States
and subject to its jurisdiction.
By itself, birth within the territorial limits of the United States, as the case of the Indians indicated, did not make one automatically "subject to the jurisdiction" of the United States. And "jurisdiction" did not mean simply subject to the laws of the United States or subject to the jurisdiction of its courts. Rather, "jurisdiction" meant exclusive "allegiance" to the United States. Not all who were subject to the laws owed allegiance to the United States.
As Senator Howard remarked, the requirement of "jurisdiction," understood in the sense of "allegiance," "will not, of course, include persons born in the United States who are foreigners, aliens, who belong to the families of ambassadors or foreign ministers accredited to the Government of the United States."
Guide to the Constitution