Why doesn't a time cutoff convert NP problems into co-NP?
The mistake I made was in the very last step: flipping the result. Doing that requires more than an $\mathsf{NP}$ machine, because I'm recombining the massively parallel computation then doing something else. $\mathsf{NP}$ machines finish with the merging, they can't continue on.

To run my algorithm I'd need a machine with the ability to query an $\mathsf{NP}$ machine then continue. A $\mathsf{P^{NP}}$ machine can do that... but it makes sense that $\mathsf{P^{NP}}$ would contain $\mathsf{coNP}$.

See also: Oracle Machines (Complexity Classes), and the Polynomial Hierarchy.