A typical counterexample is like the following. For concreteness, let's take $p=1$ and $n=2$, and $\Omega = B(0,1) \subset \mathbb{R}^2$.
Let $$f_n(t) = \begin{cases} nt, & 0 \le t \le 1/n \\ 1, & t > 1/n.\end{cases}$$
Set $v_n(x) = f_n(|x|)$, so $v_n$ is continuous and $v_n(0)=0$. (If you like you may modify the function $f_n$ slightly to make it $C^\infty$.)
You can check that $|\nabla v_n(x)| = n$ for $0 < |x| < 1/n$ and $|\nabla v_n(x)| = 0$ for $1/n < x < 1$. Thus $|v_n|_{W^{1,1}} = n \cdot m(B(0,1/n)) = \pi/n \to 0$. But $v_n \to 1$ monotonically almost everywhere, so $v_n \to 1$ in $L^1(\Omega)$.
The flaw in your proof is in looking at the properties of $v$. You defined $v$ as the $L^p$ and a.e. limit of the $v_n$, so it's only well-defined up to null sets. In general you can't choose a continuous representative for it (you claimed you could but didn't justify it), so speaking of $v(0)$ doesn't really make sense. In this example you can choose a continuous representative (namely 1) but you don't have $v_n \to v$ pointwise everywhere, so using that representative you cannot conclude $v(0)=0$.
You can find a full proof (to my knowledge the simpler one currently known) in the paper [1] and in the book [2], chapter I, §2.1 pp. 14-21. The original proof of Arthur Korn is so long and involved that K.O. Friedrichs, who gave a much simpler yet sophisticated proof, had doubts on his validity: starting from the work of Friedrichs, several authors gave their (in general quite complex) proofs, until Olga Oleĭnik gave a much shorter and simpler one (despite being still not elementary).
New edit. While ordering my library, I noted reference [1b]: in this paper Oleĭnik an Kondratiev prove the classical second Korn inequality for bounded domains satisfying the cone condition (theorem 1, a three page proof) and for certain classes of unbounded domains. They also prove that the constants in the inequality are sharp in some precise sense.
References
[1] Vladimir Alexandrovitch Kondratiev, Olga Arsenievna Oleĭnik,
"On Korn’s inequalities" (English), Comptes Rendus de l’Académie des Sciences, Série I, 308, No. 16, pp. 483-487 (1989), MR0995908, Zbl 0698.35067.
[1b] Vladimir Alexandrovitch Kondrat’ev, Olga Arsenievna Oleĭnik, "Hardy’s and Korn’s type inequalities and their applications". (English) Rendiconti di Matematica e delle sue Applicazioni, VII Serie 10, No. 3, 641-666 (1990), MR1080319, Zbl 0767.35020, also found in the commemorative book Scritti matematici. Dedicati a Maria Adelaide Sneider, Università "La Sapienza", 415-440 (1990).
[2] Olga Arsenievna Oleĭnik, Alexei Stanislavovich Shamaev, Grigorii Andronikovich Yosifian, Mathematical problems in elasticity and homogenization. (English) Studies in Mathematics and its Applications. 26. Amsterdam-London-New York-Tokyo: North- Holland, pp. xiii+398 (1992), ISBN: 0-444-88441-6, MR1195131, Zbl 0768.73003.
Best Answer
It seems to me that some useful information you can find in paragraph 1.5 of "Differentiable Functions on Bad Domains" by V. G. Mazia, S. Pobozchi.