Proving the existence of Nash Equilibrium using alternate approaches
It seems unlikely that an elementary proof exists, because the Kakutani fixed point theorem follows from the theorem proving the existence of Nash equilibria. So this is a strong theorem.



The rest of this answer has been made before the question has been edited to specifically exclude it. (After the answer was already posted and the OP read it.)



There are several ways the existence of NE's in finite games can be proven.

One can use Brouwer's fixed point theorem.

Or, one can do so using Sperner's lemma, stopping by at Brouwer's as an intermediate step. Sperner, Brouwer and Nash