Actually I think there is a subtle obstacle for which you can't appeal to Godel's 2nd incompleteness theorem and so the proof of Jech is necessary:
When we talk about $V_k$ as a model of $ZFC$ and when we state $V_k\vDash ZFC$ we are referring to the element of a model of set theory, since $V_k$ to be a model must be a set, so must be an element of some model of set theory.
In practice the statement $ZFC \vdash I$ is not equivalent to say "$ZFC$ proves it has a model" (and so "$ZFC$ proves its own consistency") but it's equivalent instead to the following: "$ZFC$ proves that, if it has a model (and thus IF it is consistent), then it has also an inner model (a second model which is an element of the previous model)". The contradiction thus rely really only on what Jech stated.
To see this more clearly, just think about the other possibility: if $ZFC$ is not consistent we would have $ZFC\vdash I$ since a non consistent theory proves everything. So it's clear that "$ZFC\vdash I$" is not equivalent to "$ZFC$ has a model".
One final remark: remember that the 2nd incompleteness theorem is to be intended more as "there is no first order logic formula that truly express the concept of being consistent" and not "a theory can not prove its own consistency".
In fact, if you read the proof of Godel's theorem the formula $Cons(T)$ is equivalent to the statement "T is consistent" only in the standard model of natural numbers. If you pick a non standard model the formula has no meaning at all (since it's not really defined for any non standard natural number which is an element of the non standard model) and that's the reason why it can be falsified.