I'd just like to point out that there is a monad on $Top$, (which in the homotopy category looks rather dull,) assigning to each space $X$ its cone $CX$, the mapping cylinder of $X\to * $. The unit map is the inclusion of $X \to CX$, and the composition $CCX\to CX$ may as well be the map $ [[x,s],t]\mapsto [x,s+t-ts].$ (This ugly formula is just a natural obfuscation of the heuristic description of $CX$ as the union of convex combinations of points $x\in X$ and the new point $*$.) Another way to think of it is that $CX$ is the underlying space of the free contraction of $X$.
The topological (realization of the) simplex category is just the orbit of a one-point space $\star$ under this monad, together with the maps derived from the monad and the maps $C\star\to \star$ and $\star\to *\to C\star$. I think this gets at what Grigory M means above by "most natural" contractible set on $n$ points. Somewhere in this nonsense I should say "Bar construction", but I can't remember precisely where.