I knew someone that unintentionally carried a switchblade in their carry-on bag to NYC, discovered that it was in their bag when they were unpacking at the hotel, and then intentionally carried it back home to Chicago in their same carry-on. Both times it passed through an x-ray machine without comment.
This shit is all security theater. Look up how often Red Teams get guns, ammunition, and simulated explosive devices past TSA; you'll be unpleasantly surprised.
So according to the TSA website you are allowed to bring nail clippers in your carry-on and check-in but they always have some bullshit text that say it is up to the agent.
I have been told by the TSA agents they only really ever take it if it has a pointed file on it, as those can be made into disguised knives. This is why you see much fewer of them on clippers these days.