What is wrong with this proof that inaccessible cardinals are inconsistent

Your claim that "is $n$-unboundedly elementary" is $\Pi_3$ for every finite $n$ (since we already run into issues when $n=2$, there's no point in going transfinite yet) is unjustified: the complexity of the definition as given increases as $n$ does. In general, writing "$E_n$" for "is $n$-unboundedly elementary," we have the suggestive relative complexity expression $$E_{n+1}=\Pi_2(\Pi_1\wedge E_n).$$ The outer $\Pi_2$ is "$\forall b<n\forall \gamma\exists \delta>\gamma\exists W$." This $W$ is implicitly invoked in your post - it's intended to be the pair $\langle V_\alpha,V_\beta\rangle$, and the "$\Pi_1$" in the parenthetical part takes care of saying this. Finally, we have the reference to $E_n$. In general, for $n>0$ the given definition of $E_n$ is $\Pi_{1+2n}$.

The crucial point is that when we invoke $E_n$ in defining $E_{n+1}$, we're doing so within the scope of some already-introduced quantifiers (the "$\Pi_2$" bit). So we don't get this for free.

Note that this is exactly the same reason that the recursive definition of truth doesn't give a contradiction to Tarski's undefinability theorem.

The recursive definition of $\eta$-unbounded elementary which you propose cannot actually be carried out in ZFC. Recall that the way recursive definitions work is that you prove by induction on $\eta$ that there is a unique function $F_\eta$ defined on $\eta$ which takes each $\alpha<\eta$ to the $\alpha$th stage of your definition. This only works if the $\alpha$th stage of your definition is a set, so that this function $F_\eta$ actually is a set (so you can quantify over such functions). In your case, you would want to be defining the class of all $\alpha$-unbounded elementary ordinals as the $\alpha$th stage, which you cannot do. (Note that for variant notions like that of $\eta$-inaccessibility, you can define the set of $\eta$-inaccessible cardinals less than $\kappa$ for any fixed $\kappa$ recursively, since $\eta$-inaccessibility of a cardinal depends only on the cardinals smaller than it. This doesn't work for $\eta$-unbounded elementary ordinals, though, since to determine whether an ordinal is $\eta$-unbounded elementary, you need to know the entire proper class of $\beta$-unbounded elementary ordinals for each $\beta<\eta$.)

Incidentally, you can carry out this definition in MK, where class quantification is allowed (and can be used in Separation, and thus in proofs by induction where you want to define "the set of counterexamples" to prove it is empty). However, your $\Sigma_4$-correctness statement then only works for statements using set quantifiers, and the recursive definition of $2$-unboundedly elementary would need to use class quantifiers (or, if you did it with set quantifiers by brute force instead of by recursion, it would need to have higher complexity, as described in Noah's answer).