Well, I've done some reading and (re)discovered an old paper of Peter Johstone which comes pretty close to answering this question using topos theory. It follows the idea I posted as a "possible lead" in my EDIT.

First some background information:

It was shown by Joyal that simplicial sets is the classifying topos for "interval objects" (this is explained for instance in "Sheaves in Geometry and Logic" by Mac Lane and Moerdijk, in which they refer to interval objects as order linear orders). By an interval object in $Set$, one roughly means a linearly ordered set together with a top and bottom element. You can say $I$ in a topos $\mathcal{E}$ is an interval object if and only if $Hom(E,I)$ is an interval object in $Set$ for all $E \in \mathcal{E}$. Since $Set^{\Delta^{op}}$ is the classifying topos for interval objects, for any topos $\mathcal{E}$, there is an equivalence of categories between the category of geometic morphisms $Hom(\mathcal{E},Set^{\Delta^{op}})$ and the category of interval objects in $\mathcal{E}$, $Int(\mathcal{E})$. Notice that the inverse image functor $f^*:Set^{\Delta^{op}} \to \mathcal{E}$ of a geometric morphism $f:\mathcal{E} \to Set^{\Delta^{op}}$ is always left-exact, so, this can be thought of as a "geometric realization functor with values in $\mathcal{E}$".

Now, the classical geometric realization functor $Set^{\Delta^{op}} \to Top$ nearly fits in this framework- it is left-exact and is uniquely determined by the fact that the universal interval object of simplicial sets is mapped to the standard unit interval $[0,1]$. However, $Top$ is not a topos. This is where Peter Johstone's 1977 Paper "On a topological topos" comes in. In this paper he constructs a topos $\mathcal{T}$ which contains sequential topological spaces (and hence e.g. CW-complexes) as a reflective subcategory. (In case you are interested, this topos is the topos of sheaves with respect to the canonical topology on the fullsubcategory of $Top$ consisting of the one-point space and the one-point compactification of $\mathbb{N}$.) Moreover, the inclusion of sequential spaces into $\mathcal{T}$ preserves lots of colimits- e.g. all colimits you'd use to construct CW-complexes. Now, since the standard unit interval $[0,1]$ is an object of $\mathcal{T}$, it corresponds to a unique geometric morphism $r:\mathcal{T} \to Set^{\Delta^{op}}$. Johnstone then proves that if $X$ is a simplicial set, then $r^*(X)$ is exactly $|X|$ (as a sequential space considered as an object of $\mathcal{T}$) AND that if $T \in \mathcal{T}$ is a sequential space, then $r_*(T) \cong Sing(T)$.

This is somewhat satisfying. However, for it to truly be satisfying, we'd have to either make sense out of why $\mathcal{T}$ is a natural choice, or, show that any "suitable choice" of a topos would give the same answer. Moreover, although intuively somehow clear, I would like to make sense out of in what way the "standard unit interval" $[0,1]$ is really a "canonical interval object".

Well, I've done some reading and (re)discovered an old paper of Peter Johstone which comes pretty close to answering this question using topos theory. It follows the idea I posted as a "possible lead" in my EDIT.

First some background information:

It was shown by Joyal that simplicial sets is the classifying topos for "interval objects" (this is explained for instance in "Sheaves in Geometry and Logic" by Mac Lane and Moerdijk, in which they refer to interval objects as linear orders). By an interval object in $Set$, one roughly means a linearly ordered set together with a top and bottom element. You can say $I$ in a topos $\mathcal{E}$ is an interval object if and only if $Hom(E,I)$ is an interval object in $Set$ for all $E \in \mathcal{E}$. Since $Set^{\Delta^{op}}$ is the classifying topos for interval objects, for any topos $\mathcal{E}$, there is an equivalence of categories between the category of geometic morphisms $Hom(\mathcal{E},Set^{\Delta^{op}})$ and the category of interval objects in $\mathcal{E}$, $Int(\mathcal{E})$. Notice that the inverse image functor $f^*:Set^{\Delta^{op}} \to \mathcal{E}$ of a geometric morphism $f:\mathcal{E} \to Set^{\Delta^{op}}$ is always left-exact, so, this can be thought of as a "geometric realization functor with values in $\mathcal{E}$".

Now, the classical geometric realization functor $Set^{\Delta^{op}} \to Top$ nearly fits in this framework- it is left-exact and is uniquely determined by the fact that the universal interval object of simplicial sets is mapped to the standard unit interval $[0,1]$. However, $Top$ is not a topos. This is where Peter Johstone's 1977 Paper "On a topological topos" comes in. In this paper he constructs a topos $\mathcal{T}$ which contains sequential topological spaces (and hence e.g. CW-complexes) as a reflective subcategory. (In case you are interested, this topos is the topos of sheaves with respect to the canonical topology on the fullsubcategory of $Top$ consisting of the one-point space and the one-point compactification of $\mathbb{N}$.) Moreover, the inclusion of sequential spaces into $\mathcal{T}$ preserves lots of colimits- e.g. all colimits you'd use to construct CW-complexes. Now, since the standard unit interval $[0,1]$ is an object of $\mathcal{T}$, it corresponds to a unique geometric morphism $r:\mathcal{T} \to Set^{\Delta^{op}}$. Johnstone then proves that if $X$ is a simplicial set, then $r^*(X)$ is exactly $|X|$ (as a sequential space considered as an object of $\mathcal{T}$) AND that if $T \in \mathcal{T}$ is a sequential space, then $r_*(T) \cong Sing(T)$.

This is somewhat satisfying. However, for it to truly be satisfying, we'd have to either make sense out of why $\mathcal{T}$ is a natural choice, or, show that any "suitable choice" of a topos would give the same answer. Moreover, although intuively somehow clear, I would like to make sense out of in what way the "standard unit interval" $[0,1]$ is really a "canonical interval object".