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.