We know that $E^\circ_{\text{cell}}=E^\circ_{\text{reduction at cathode}}+E^\circ_{\text{oxidation at anode}}$. For example, in a cell $\ce{Zn(s) | Zn^2+(aq) || Cu^2+(aq) | Cu(s)}$, $$E^\circ_{\text{cell}}=E^\circ_{\text{reduction at cathode}}+E^\circ_{\text{oxidation at anode}}=E^\circ_{\ce{Cu^2+/Cu}}+E^\circ_{\ce{Zn/Zn^2+}}$$
This makes sense to me. I think of $E^\circ_{\text{cell}}$ as the $E^\circ$ value of the net cell reaction i.e. of $\ce{Zn(s) + Cu^2+(aq) -> Zn^2+(aq) + Cu(s)}$.
But, consider this cell: $\ce{Zn(s) | Zn^2+(aq) || Ag^+(aq) | Ag(s)}$ According to my textbook, the $E^\circ_{\text{cell}}$ value here is again defined by:
$$E^\circ_{\text{cell}}=E^\circ_{\text{reduction at cathode}}+E^\circ_{\text{oxidation at anode}}=E^\circ_{\ce{Ag+/Ag}}+E^\circ_{\ce{Zn/Zn^2+}}$$
But, I think this is not correct.
Consider the fact that this cell is made up of two half cell reactions:
$$ \begin{array}{} \text{Oxidation}&\ce{Zn(s)}&\ce{-> Zn^2+(aq) + 2e-}&E_1^\circ\\ \text{Reduction}&\ce{Ag^+(aq) + e-}&\ce{-> Ag(s)}&E_2^\circ\\ \text{Net cell reaction}&\ce{Zn(s) + 2Ag^+(aq)}&\ce{-> Zn^2+(aq) + 2Ag(s)}&E_3^\circ\\ \end{array} $$
and that we can't add $E^\circ$ values directly, since they are an intensive property. Instead, we need to say that $\Delta G_3=\Delta G_1+\Delta G_2$ and then write $2E_3^\circ=2E_2^\circ+E_1^\circ$$2E_3^\circ=2E_1^\circ+E_2^\circ$ or $E_3^\circ=\frac{2E_2^\circ+E_1^\circ}2$$E_3^\circ=\frac{2E_1^\circ+E_2^\circ}2$, which is definitely $\neq E_2^\circ+E_1^\circ$$\neq E_1^\circ+E_2^\circ$, as my textbook says.
I believe I have correctly presented everything here, and cannot figure out my mistake. Where am I wrong?