This seems to be false.
Let $n=6$, $\ p=1/2$,$$A_0 = \left(\begin{matrix}0 & 1 \\0 & 1 \\\end{matrix}\right),\ \A_1 = \left(\begin{matrix}1 & 0 \\0 & 0 \\\end{matrix}\right).$$
So the norm of the product is either $\sqrt{2}$, 1 or 0. (If the norms or matrices seem artificial, they can be adjusted by $\epsilon$ without affecting the result.)
The two expectations are $(720+127\sqrt{2})/4096 = .2196$ for the uncorrelated case and$(724+126\sqrt{2})/4096 = .2203$ for the correlated case.
