Here are some crazy ideas. I am posting now just to get the ball rolling, and if anyone (myself included) comes up with anything of substance in this direction you should just edit this answer. I am making it community wiki for ease of editing, and because the ideas are currently just kind of wonky.
I think recovering the standard geometric realization functor from pure abstract nonsense might be hard, just because there would have to be an implicit abstract nonsense construction of the closed interval as a topological space out of just finite linear orders, and I feel I would have seen that somewhere before.
On the other hand, I recently learned that every finite CW complex is weakly equivalent to a finite topological space. For example the circle is weakly equivalent to a topological space with 4 points. In fact "for any finite abstract simplicial complex K, there is a finite topological space $X_K$ and a weak homotopy equivalence $f : |K| \to X_K$ where $|K|$ is the geometric realization of $K$." (according to wikipedia). So maybe we could make this construction functorial from finite simplicial complexes to finite topological spaces. Maybe this functor (which factors through geometric realization, and is "just as good" as far as algebraic topology is concerned) could then be extended to simplicial sets. Since the construction should be more combinatorial, and not involve the reals in any way, I feel like this new functor (if it exists) might be more amenable to an "abstract nonsense" description. As far as algebraic topology is concerned, this new functor might be "just as good" as geometric realization.
I have some ideas for what this functor might look like, but I am still playing around with small examples. Feel free to join in the madness if you like, and add you edits with your name attached if you like.