Consistency is a $\Pi_1$ statement, and in fact a universal $\Pi_1$ statement: if we have a strong enough base theory (say $\mathsf{I}\Delta_0$) then adding the consistency of $\mathsf{ZFC}$ will allow us to obtain all $\Pi_1$ consequences of $\mathsf{ZFC}$ but nothing more, e.g. we cannot prove existence of any new object.
For the first one, we need to show that we can define the truth of $\Pi_1$-formulas, and that consistency of $\mathsf{ZFC}$ is equivalent to $\Pi_1$-soundness of $\mathsf{ZFC}$ (this follows from $\mathsf{ZFC}$ being $\Sigma_1$-complete). Then if $\pi$ is a proof of $\mathsf{ZFC} \vdash \varphi$, then it is a finite object and therefore we can show its existence in $T$, and by $\Pi_1$-soundness it would follow $Tr(\varphi)$. Then we have to show that we can derive $\varphi$ from $Tr(\varphi)$ (which is proven by induction over the structure of the formulas).
For the second one, note that adding $\Pi_1$ axioms doesn't imply existence of any new objects.