(After Noah Stein's answer) The following counter-example is due to Hall. The $5\times5$ symmetric matrix $$S=\begin{pmatrix} 4 & 0 & 0 & 2 & 2 \\\\ 0 & 4 & 3 & 0 & 2 \\\\ 0 & 3 & 4 & 2 & 0 \\\\ 2 & 0 & 2 & 4 & 0 \\\\ 2 & 2 & 0 & 0 & 4 \end{pmatrix}$$ has non-negative entries and is positive semi-definite. Yet, it cannot be written as the sum of $v_j\otimes v_j$ where the vectors $v_j$ are non-negative. By duality, this proves that $C$ is not the convex hull of $S\cup P$.

The example is analyzed in details in Exercise 347 of my list.

(After Noah Stein's answer) By definition, the dual cone $C^\star$ is spanned by matrices $v\otimes v$ with $v\ge0$. The following counter-example is due to Hall. The $5\times5$ symmetric matrix $$S=\begin{pmatrix} 4 & 0 & 0 & 2 & 2 \\\\ 0 & 4 & 3 & 0 & 2 \\\\ 0 & 3 & 4 & 2 & 0 \\\\ 2 & 0 & 2 & 4 & 0 \\\\ 2 & 2 & 0 & 0 & 4 \end{pmatrix}$$ has non-negative entries and is positive semi-definite. Therefore, it belongs to $(S\cup P)^\star$. Yet, it cannot be written as the sum of $v_j\otimes v_j$ where the vectors $v_j$ are non-negative. Therefore $A\not\in C^\star$. By duality, this proves that $C$ is not the convex hull of $S\cup P$.

The example is analyzed in details in Exercise 347 of my list.