Let $\Omega$ be an open subset of $\mathbb{R}^N$. The proof that the Sobolev space $W^{1,p}(\Omega)$ is a separable space, when $1\leq p<\infty$, is usually done by defining an isometry between $W^{1,p}(\Omega)$ and $L^p(\Omega)\times L^p(\Omega)^N$. I completely understand the proof (see for example Brezis' book - Functional Analysis, Sobolev Spaces and Partial Differential Equations for a reference or this link: $W^{1,p}$ is separable for $1\leq p<\infty$).
We know that any subset of a separable metric space is separable as well as any linear subspace of a separable Banach space is separable.
My question is: Why cannot we say, directly, that $W^{1,p}(\Omega)$ is a separable space as a linear subspace of the separable space $L^p(\Omega)$? Is there any problem with this reasoning or, maybe, the usual proof is just done in order to learn a new technique in the work with these spaces?