Is the zero vector not being a possible eigenvector just an arbitrary decision? I understand that if 0 is allowed to be an eigenvector, then it would be an eigenvector of every matrix (corresponding to some λ).  

But why is that a bad thing?  Does allowing this cause any contradiction? Or this is really just some convention for convenience? In that case, what is this convenience?  

Thanks.
[Made my comment an answer]

It's just not useful. Eigenvectors tell you something about the operator (many times, they tell you everything). The zero vector tells you absolutely nothing about it - not even an eigenvalue, since $L0=\lambda 0$ for any $\lambda$ and any $L$.
Because it would be an eigenvector corresponding to EVERY $\lambda$, so the notion of eigenvalue would be quite useless if every matrix could have any possible $\lambda$ as eigenvalue.