Which is more natural?
- We're getting a lot of new principals who, quite frankly, are not very skillful at handling student discipline.
- We're getting a lot of new principals who are, quite frankly, not very skillful at handling student discipline.
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Both are natural; both would be used, and both would be understood.
Both seem fine to me. All you are doing is changing where the linking verb "are" is! Both are natural to me too.
The linking verb can be moved around due to:
A linking verb connects the subject of a sentence with a word that gives information about the subject, such as a condition or relationship. They do not show any action; they simply link the subject with the rest of the sentence. Dictionary
Additionally, it is an auxiliary verb, a form of "be", and in this case:
Auxiliary verbs always go before main verbs. Source
Therefore the "are" is always before the main verb "handling", so the meaning doesn't change.
Agreeing with other answers, giving some of my own exposition:
By analogy:
I have a dog who, quite frankly, eats cat poop.
I have a dog who eats, quite frankly, cat poop.
The part after "quite frankly" is what you are being frank about, and perhaps you consider it a sensitive or off-color subject. So in the first case, you're sensitive that your dog eats cat poop -- a nasty thing he is prone to do -- while in the second case, you're sensitive that cat poop is what he eats -- a nasty thing he eats, you expect him to eat something but not that. These are very, very close in meaning.
In your original sentence pair, it's even less of a distinction, because the verb is a verb of being and not action. You're sensitive about either "they are-not-skillful" -- an unfortunate combined condition of being -- or else that they are not-skillful -- an unfortunate attribute that they present. These are also extremely close in meaning.
Most listeners won't even think this hard about what you're saying. Go ahead and use whichever you want; both are acceptable.
I agree with the above answers that both are fine. That said, if I were to nitpick, I like the first option slightly more.