I am asking about the answer given by @aws on the page Why do class-sized models escape the completeness theorem?, and whether and why what aws says is correct. In the answer aws sketches a fake "proof" that ZFC proves its own consistency, by defining a class function $F$ that determines truth given a string/godel number associated to an input formula. They say that this proof goes wrong because "We want $F(\forall x \phi(x))$ to be one if $F(\phi(a))=1$ for every $a\in V$ and zero otherwise. However, we can't state this directly in the language of set theory," and "because the formulas can have parameters from V, there is a proper class of them, and in particular the inductive clauses for quantifiers would be most problematic." However, I don't see why it isn't perfectly fine to define $$ F(\ulcorner\forall x \phi \urcorner,[a])= \begin{cases} 1 & \forall x (F(\ulcorner\phi \urcorner, [a^i_x])=1)\\ 0 & \textrm{else} \end{cases} $$ and then use recursion to find a class function/truth formula $F$, proving the consistency of ZFC.
Am I making a mistake? Or am I right and the problem with this construction lies elsewhere (like maybe in the fact that resulting class function $F$ will have free parameters).