Mutiny is likely. American officers in particular will tend not to follow orders to commit an obvious war crime. This isn't an extralegal check , it's very much legal.
But the system will fail if "the whole chain is stupid." This is why it's so dangerous to have a President obsessed with loyalty.
Yes, and that's why we have so many obligatory legal oaths. It is kind of astonishing that Americans without the internet developed a pretty rigorous system so long ago - Joe Rogan (sorry) summarized it well when he said that smart people think they're smart, but they're actually just regurgitating things they've learned from truly genius people from past generations
What makes you think mutiny would be likely? It would depend on many factors such as the psychological makeup of the launch & command staff and the circumstances du jour. No doubt they made sure independent-minded soldiers likely to be unreliable were not placed anywhere near any silo.
But the system will fail if "the whole chain is stupid." This is why it's so dangerous to have a President obsessed with loyalty.