The parenthetical phrase means that the phrase is to be considered separate from the sentence. That’s important for some to note because it helps some folks see how to set up the sentence (and isolate the phrase) and not try to connect said phrase artificially with the sentence into which it has been inserted.
You are correct. It might be more precise to state that the parenthetical phrase is a full clause within the sentence because the fact of its being parenthetical defines it as indeed being “connected” to the larger phrase (sentence) which determines the pronoun. “Who” (in this case referencing the subject “Hillary”) is, as you say, the only possible pronoun; but that’s because it’s the subject of the main verb, not because it happens to occur in a parenthetical phrase. Thanks!