How to define the natural rate output
That is not correct  conclusion and there are several problems with the formulation you use.

$A$ is not a shock - it is a technology parameter of production function (see Woodford Interest and Prices pp 148). Heck, $A$ cannot even be zero. Following Woodford:


$A_t&gt; 0$ is a time-varying exogenous technology factor

there can be shocks to $A_t$ that increase it or decrease it but $A_t$ itself is strictly positive at all times.

It is actually not $s(Y^n,Y^n,ξ)=μ^{−1}$ but $s(Y^n_t,Y^n_t;ξ)$. This might seem trivial but note $s(.)$ has only 2 arguments ($Y^n_t$ and $Y^n_t;\xi$  not 3 arguments $Y^n_t$, $Y^n_t$ and $\xi$). In addition, the solution to $s(Y^n,Y^n,ξ)=μ^{−1}$ gives you the natural rate of output not the equation itself. In this context how I read the Woodford semicolon (;)  is used to separate variables and shock parameters. Hence, it is just to indicate that the ξ is not a variable but given by the shock parameters of the economy (i.e. indicate variable conditional on vector of shocks.

Your definition of utility function is extremely non-standard, you define it as:


$$ u= \frac{\xi_t C_t^{1-\sigma}}{1-\sigma} $$
but why are the shocks $\xi_t$ even multiplied by consumption in the first place? In Woodford the equation (1.1) is given as:
$$E_0 \left[ \sum^{\infty}_{t=0} \beta^t \left( u_t(C_t, M_t/P_t; \xi_t) - \int^{1}_{0} v(h_t(i);\xi_t)di \right) \right]$$

first as you can see $C_t$ is not even conditional on shocks $\xi_t$, the real money holding is $M_t/P_t; \xi_t$ is conditional on vector of shocks.
you can't just assume $u_t(C_t, M_t/P_t; \xi_t) = \xi_t C_t^{1-\sigma}$, I mean that makes no economic sense whatsoever, why should utility be product of vector of shocks and consumption?

I don't think you will get the same result with some more reasonable utility function and take into account that the variables are conditional on the vector of shocks (e.g. $ M_t/P_t; \xi_t$ does not mean  $M_t/P_t$ has mean zero even if $\xi_t$ is given by Gaussian distribution with mean zero).