P&S on page 213 arrive at the following identity for the two-point function (eq. 7.5)
$$\langle \Omega|\phi(x)\phi(y)|\Omega\rangle={\sum_{\lambda} \int \dfrac{d^4p}{(2\pi)^4}\dfrac{ie^{-ip.(x-y)}}{p^2-m_{\lambda}^2+i\epsilon}}|\langle \Omega|\phi(0)|\lambda_0\rangle|^2.\tag{7.5}$$
This is done by using the following resolution of the identity $$ \textbf{1}=|\Omega\rangle \langle \Omega| + {\sum_{\lambda}} \int \dfrac{d^3p}{(2\pi)^3} \dfrac{1}{2E_p(\lambda)} |\lambda_p\rangle \langle \lambda_p|.\tag{7.2}$$
If one follows the derivation in P&S then I don't see why eq. 7.5 shouldn't hold for a two-point function consisting of any two observables $O(x)O(y)$. the singular pole structure of the two-point function actually comes from the above resolution of the identity and is not inherent to the two-point $\phi$ function. In other words, I'm saying that the following should hold as well:
$$\langle \Omega|O(x)O(y)|\Omega\rangle={\sum_{\lambda} \int \dfrac{d^4p}{(2\pi)^4}\dfrac{ie^{-ip.(x-y)}}{p^2-m_{\lambda}^2+i\epsilon}}|\langle \Omega|O(0)|\lambda_0\rangle|^2.$$
In particular this should hold for, say, $O(x)=\pi(x)$, the variable conjugate to $\phi$. Hence I can also conclude that the LSZ reduction formula is also valid for any type of observables (i.e. we can relate the $S$-matrix elements to the vacuum expectation value of any kind of observables). So my question would be: Am I correct in this regard or am I missing something?
Of course I suspect why this is done particularly for $\phi$ two-point/$n$-point function (for LSZ) is that we can then directly relate the $S$-matrix elements to the fully connected amputated Feynman diagrams via Gell-Mann Low Theorem.
