P=NP giving a deterministic algorithm for SAT
I like the way you think, but there's a gap in your attempt.

$\mathrm{P}$ and&nbsp;$\mathrm{NP}$ are classes of decision problems, i.e., computational tasks where the answer is either "yes" or "no". Nondeterministically generating a satisfying assignment isn't a decision problem, so it isn't in&nbsp;$\mathrm{NP}$, so the assumption that $\mathrm{P}=\mathrm{NP}$ doesn't give you a deterministic algorithm for it. It feels like it ought to, but that's not a proof.