I'm not sure if we need a 4th proof of this fact, but after wondering about this for several years I realized it can be proven in a very formulaic manner: turn everything in sight into (the classifying space of) a category and apply Quillen's Theorem A. Let's see how it goes.
The simplicial complex $|K|$ is more-or-less already a category - if we think of $K$ as just its set of simplices, then it's partially ordered by inclusion, and the geometric realization of this poset $(K, \subseteq)$ is the usual barycentric subdivision of $|K|$ (and in particular, it's homeomorphic to $|K|$).
Next, every simplicial $X$ set can be ``turned into a category" by taking the category of simplices $\textrm{Simp} (X)$. I don't know a simple proof that $|\textrm{Simp} (X)|$ is homotopy equivalent to $|X|$, but it is proven in Hirschhorn's book Model Categories and Their Localizations, Theorem 18.9.3.
We want a functor between $\textrm{Simp} (X(K))$ and $(K, \subseteq)$. This is straightforward; objects of $\textrm{Simp} (X(K))$ are (in bijection with) simplices of $X(K)$, and we can send a list $(x_0, \ldots, x_n)$ to the simplex $\{x_0, \ldots, x_n\}$ in $K$. All diagrams in $K$ commute (since it's a poset), and this makes it easy to verify that this defines a covariant functor $s:\, \textrm{Simp} (X(K))\rightarrow (K, \subseteq)$.
Finally, let's try applying Theorem A and see what happens. Fix a simplex $\sigma\in K$. The fiber of $s$ consisting of all simplices in $X(K)$ that map to faces of $\sigma$ can be though of as the category of lists in the set $\sigma$, with a morphism of lists being a way of embedding one list as a sublist of another. It is more-or-less immediate from the definitions that this fiber category is isomorphic to $\textrm{Simp} (N_* (I(\sigma)))$, where $I (\sigma)$ is the indiscrete category on the set $\sigma$ (that is, the object set of $I(\sigma)$ is $\sigma$, and each morphism set has exactly one element). Since $I(\sigma)$ is equivalent to the trivial category, $|N_* (I(\sigma))|$ is contractible. By the discussion above, so is $|\textrm{Simp} (N_* (I(\sigma)))|$, and Theorem A says that $|s|:\, |\textrm{Simp} X(K)|\stackrel{\simeq}{\rightarrow} |K|$ is a homotopy equivalence.