Finding the mistake(s) within this "proof" of NP being closed for complement
You just need to show that swapping final and non-final states doesn't complement the language accepted by an NTM, e.g., by showing that, for at least one NTM&nbsp;$N$ (which you can construct to have whatever properties you need), there is a  string that is accepted by both $N$ and $N'$, or rejected by both.