You can prove the Birkhoff-von Neumann theorem directly with linear programming. Depending on your taste it is a quite elegant way to prove that result. There are basically two ways - one to use the conditions for a vertex of a polytope given by constraints to show that a doubly stochastic matrix which is a vertex of the Birkhoff polytope must have a row or column with only one nonzero entry, then induce. This does not use the full "fundamental theorem of linear programming".

The other approach is to observe that at a vertex there is a full dimensional set of linear objectives for which the vertex is optimal, formulate the dual program and then show that the 2n unconstrained dual variables lie on an n dimensional space; complementary slackness then shows that the primal variable has only n nonzero elements, double stochasticity then guarantees there must be one in each row, one in each column, and each must be unity - therefore a permutation matrix. Obviously this approach really does exploit the linear program structure, if that is what you want to teach.

I came up with this myself so don't know of an actual reference, but it should not be that novel.

You can also prove Birkhoff-von Neumann are a max flow/min cut theorem (which is pretty well known) but I do not find that as elegant. However if you are emphasizing max flow/min cut as opposed to the linear programming structure, then you might want to do that one.