This is an interesting idea, but I don't believe that your argument succeeds.
Notice first that if it were correct, then it would also refute the existence of $I_1$ rank-into-rank cardinals, that is, $j:V_{\lambda+1}\to V_{\lambda+1}$, since $\lambda=\kappa_\omega$ and the singularity of this cardinal is revealed in $V_{\lambda+1}$.
Some people might object that you haven't said what you mean by iterating $j$, but this is not actually a problem, for there is a natural way to do this. Any class embedding $j:V\to M$ can be iterated, since the definition of $j$ can be extended to any proper class $A$ by defining $j(A)=\bigcup_\alpha j(A\cap V_\alpha)$. This way, we get $j(j)$, which will be an elementary embedding from $M$ to $j(M)$. Thus, if we start with $j:V\to V$, we may indeed iterate $j$ to form a system $V\to V\to V\to\cdots$ as you describe.
The problem is that you claimed that the direct limit of this system is $V$, but I don't see why this should be true; this seems to be a gap in your proof. It may seem reasonable to suppose that the direct limit of a system of structures, all of which are the same, to be again that same structure, but in fact there are counterexamples to this general principle. (For a quick example, embed the discrete order $\mathbb{Z}$ into itself by stretching by a factor of two; the direct limit of the iteration of this process is the dense order $\mathbb{Q}$.)
Indeed, since for the Kunen inconsistency we do not want to assume that the embedding $j$ is definable, but rather merely that it is a class in the sense of Gödel-Bernays set theory, I don't see even that the usual arguments show that the direct limit is well-founded.