There's a slightly different equivalence that is also useful. Consider the quiver with n elements, and an arrow E_i from i to i+1 and another F_i from i+1 to i for all i. The relations are then that E_i F_i = e_i and F_i E_i = e_{i+1}, where e_j is the j-th simple idempotent. This gets the same path algebra with fewer arrows and relations, but it has even less symmetry than your presentation.
A first answer to your question is that this perspective can often be useful. The reason I say this is because this perspective allows you to realize a lot of other quivers as subalgebras of matrices, and vice versa (for instance, the Borel subalgebra as the path algebra of a subquiver). It's not an extremely useful proving technique, but it can be a good way to produce a lot of quivers, especially when first learning about them.
Is it interesting? That's another question entirely. It's unfortunate that it picks out a basis in a necessary way, and so the GL_n action on M_n doesn't seem natural. I think the fact the the presentation I mention above is close to what is called a 'double quiver' is somewhat interesting. Especially if you like to think of a semisimple Lie algebra as something like the tangent bundle to the space of Borel subalgebras. Precisely, I mean that BB localization relates certain modules of g to D-modules on the space of Borel subalgebras, and so it is interesting to think of M_n as a deformation of the tangent bundle to U_n, the upper triangular matrices.
