> But only native-born US citizens are eligible for some jobs, e.g. in the military.
There are exactly two jobs where that is required, and only one is kind-of in the military, in that it is legally the apex of all military chains of command.
(There are other security-related, especially in the military, jobs where actual or potential dual citizens, the latter more commonly being an issue when people have foreign born parents, may be required to renounce any other citizenship than American.)
It is absolutely discrimination, but legal (and legally-mandatory) discrimination. That it may be “common sense” does not change that it is, absolutely, discrimination.