This ordinal is not as large as you claim. In particular, the constructible universe up to this ordinal does not satisfy ZFC minus power set, and the ordinal is not even admissible.

To see this, let $\gamma_n$ be the least $\Sigma_n$-admissible ordinal (which I understand you to mean that $L_{\gamma_n}$ satisfies KP plus $\Sigma_n$-replacement, or equivalently, $\Sigma_n$-collection); and then the ordinal in question is $\gamma=\sup_n\gamma_n$ and the corresponding structure $L_\gamma$. But this structure does not satisfy KP, since the map $n\mapsto\gamma_n$ on domain $n\in \omega$ is $\Sigma_1$-definable in $L_\gamma$. The point is that whether some $L_\beta$ satisfies a certain sentence is a $\Delta_0$ espressible assertion about $L_\beta$, since all quantifiers are bounded by $L_\beta$, and whether $\beta=\gamma_n$ is a $\Sigma_1$-property of $\beta$ inside $L_\gamma$. Namely, $\beta=\gamma_n$ if and only if there is a transitive set $X$ that thinks it is $L_\beta$ and there is a satisfaction predicate on this set fulfilling Tarski's recursive definition of truth and according to this satisfaction predicate, the set $X$ satisfies KP and every instance of $\Sigma_n$-collection, and finally no smaller $\beta'\lt\beta$ satisfies all those.