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 is 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,
is 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, let me argue directly that the direct limit is definitely
not $V$. The reason is that the direct limit model has only
$\lambda$ many subsets of $\lambda$, where
$\lambda=\kappa_\omega$. This is simply because every subset of
$\lambda$ in the direct limit is born at some stage $M_n$ as a
subset of $\kappa_n$, having the form $j_{n,\omega}(A)$ for some
$A\subset\kappa_n$. But there are only $\lambda$ many $A$ like
that, and so the limit model is missing most subsets of $\lambda$.

Note that the limit of the iteration is well-founded, by
essentially the usual Kunen argument. Namely, if it were
ill-founded, then we may consider the least $\gamma$ with
$j_{0,\omega}(\gamma)$ in the ill-founded part of the limit. By
pushing $\gamma$ into $M_n$, we may conclude on the one hand that
$j_{0,n}(\gamma)$ must be least sent into the ill-founded part,
but on the other hand we may also drop down to smaller ordinal
born at that stage still in the ill-founded part, making a
contradiction. (To formalize this precisely, one may avoid the
complications caused by $j$ being a proper class simply by
chopping off $j$ to the set $j\upharpoonright V_\alpha$, such that
the iteration whose iteration is ill-founded.) This argument relies fundamentally on the fact that the iteration of $j$ in $M_0$ produces the same limit as the iteration of $j(j)$ in $M_1$.

Finally, if on the other hand you had formed your iteration by using the same
embedding $j_{n,n+1}=j$ at each stage, then this is clearly
ill-founded, since the threads starting with $\kappa$ at each
stage form a descending sequence of ordinals in the limit.