Proof of Co-Problem being in NP if Problem is in NP using negated output
A problem is in NP if for all instances where the answer is “Yes” you can guess a hint and verify it in polynomial time. Nothing at all is said for instances where the answer is “No”. That distinction between “Yes” and “No” answers is essential.
What your proof attempt shows is that if a problem X is in NP, then the co-problem “Does X have the answer ‘No’” is in co-NP, as we would expect, and says nothing about NP.