Idea behind $\mathsf{NP}\subseteq\mathsf{P}/\mathsf{Poly}\implies\mathsf{P}=\mathsf{NP}$ not true?
Assume $\mathsf{NP}\subseteq \mathsf{P}/\mathsf{Poly}$. Let $\Pi$ be a NP-complete problem, by assumption, this problem can be solved by some circuit family. It is true that the problem $\hat{\Pi}$ where the input is a pair consisting of input to $\Pi$ and a circuit making an appropriate computation, is in $\mathsf{P}$. However, from this fact you cannot conclude that the original problem $\Pi$ is in $\mathsf{P}$. To obtain a solution to $\Pi$ using $\hat{\Pi}$ you have to guess the circuit - and this is not a priori possible in $\mathsf{P}$. You could guess the circuit nondeterministically, but then the argument would prove that $\mathsf{NP} = \mathsf{NP}$. However, if you start with a $\Sigma_2$ set instead, it is possible to fix this reasoning and obtain $\Sigma_2=\Pi_2$ (Karp-Lipton theorem).