It wasn’t the subject of the Amendment. Mere citizenship was.
The drafters of the 14th were not trying to re-write the well-understood qualifications for the Presidency. They were trying to ensure citizenship for the descendants of slaves without granting it to indians.
And what other rights have the SCOTUS seen in the 14th Amendment?
Take a look at Plyler and how the Court granted Constitutional protections to illegal aliens under the Equal Protection Clause.