Yes, that is true. But note that in its nature statements like $\operatorname{Con}(T)$ are meta-theoretic statements. So when we say that $V$ is a model of $\sf ZF$, we mean that in the meta-theory it is a model of $\sf ZF$.
This is often something which is not stressed enough in introductions to $V$ and relative consistency results: when we prove that $L$ is a model of $\sf ZFC$, we do not "just prove a meta-theoretic result", we in fact prove a stronger statement:
There is a formula $L$ in the language of set theory which defines a class that is provably transitive and contains all the ordinals, and for every axiom $\varphi$ of $\sf ZFC$, $\sf ZF\vdash\varphi^\it L$.
So not only you have this model, but in fact $\sf ZF$ itself prove that each axiom of $\sf ZFC$ holds in $L$.
Let me also share, in my first course on axiomatic set theory, which was given by the late Mati Rubin, we had proved that $\sf ZF-Reg$ and $\sf ZF$ are equiconsistent by practically proving that $\sf PRA$ proves that if there is a contradiction in $\sf ZF$, then there is one in $\sf ZF-Reg$.
Of course, the same can be done with $\sf ZF$ and $\sf ZFC$. And it is much more annoying than using the model theoretic approach. Sometimes with impunity when it comes to class models.