There are several people here much more qualified to speak about that, so I shall just give you some pointers now. One of the questions Grothendieck tried to answer when writing "Pursuing Stacks" was — I don't know how he put it, though — "what are the properties of the simplicial category which make it so useful in homotopy theory?" That is where the theory of *test categories* stems from. As Georges Maltsiniotis puts it: "Le slogan de Grothendieck est que toute catégorie test est aussi “bonne” que celle des ensembles simpliciaux pour
“faire de l’homotopie”." Which means "Grothendieck's motto is that any test category is as "good" as the category of simplicial sets to "make homotopy theory"." The theory was further developed by Denis-Charles Cisinski. The two books to read on this subject are:

Maltsiniotis's "La Théorie de l'homotopie de Grothendieck" ("Grothendieck's Homotopy Theory"), the introduction of which is remarkably well-written:

http://www.math.jussieu.fr/~maltsin/ps/prstnew.pdf

and

Cisinski's (augmented version of his) thesis "Les Préfaisceaux comme modèles des types d'homotopie" ("Presheaves as Models for Homotopy Types"):
http://www.math.univ-toulouse.fr/~dcisinsk/ast.pdf

Both are available in SMF's collection Astérisque.

I shall give you more details if nobody else shows up to explain the yoga (I myself have but a smattering of it).

EDIT: Well, here are some details. You are asking: "What is so wonderful about Δ that allows a model structure (and one, moreover, Quillen equivalent to topological spaces) appear?" The shortest answer would be: "$\Delta$ is a test category". Let's try to see what it means. (I am feeling a bit guilty, for what follows is essentially a rephrasing, with the same notations, of some parts of Maltsiniotis's crystal-clear introduction to his book. I hope it will at least benefit those who cannot read French. Please note that Maltsiniotis's book is based on material written by Grothendieck in "Pursuing Stacks" almost thirty years ago.)

The starting point of the theory of test categorie is similar to your question. Namely, Grothendieck seeks to find *all* the couples $(M, W)$ where $M$ is a category and $W \subseteq Ar(M)$ such that the localized category $W^{-1}M$ be equivalent to the homotopy category $Hot$, and such that $W$ is natural in some sense (with respect to the structure of the underlying category). Given the difficulty to answer such general a question, Grothendieck then requires of $M$ to be a presheaf category on a small category $A$. Adding another slight condition on the small category $A$ (requiring that the "nerve functor" $i_{A}^{*} : Cat \to \widehat{A}$, $C \to (a \mapsto Hom_{Cat}(A/a, C))$, send weak equivalences to weak equivalences, where weak equivalences of $Cat$ are those functors the classical nerve of which are simplicial weak equivalences, and weak equivalences in the presheaf category $\widehat{A}$ are those morphisms sent to weak equivalences of $Cat$ by the functor $A/?$), he is lead to define the notion of *weak test category*. One of the properties of such a category $A$ is that the localization of its presheaf category by weak equivalences is equivalent to the homotopy category $Hot$. Of course, the simplicial category is a test category. But it is even better that that. It is a *strict test category*, which implies (by definition) for instance that cartesian product reflects the product of homotopy types. This theory shows, by the way, that the cubical category differs from the simplicial category in this respect: indeed, the cubical category is *not* a strict test category (but it is a test category, which of course lies somewhere between being weak test and being strict test). You might think that, since the cubical category is not a strict test category, strict test categories ought to be pretty scarce. In fact, there are plenty of them. For instance, every full subcategory of $Cat$ the objects of which are non-empty, and which is stable under finite products, and one object of which has at least two objects (possibly isomorphic) is a strict test category. There are results allowing one to check that a given category is a (weak, local, strict…) test category, which I will not state here. Just one example: Joyal's category $\Theta$ (related to infinity stuff) is a test category (this was proved by Cisinski/Maltsiniotis and Ara).

Actually, there is more than that in the theory. You can ask what are the formal properties of weak equivalences of $Cat$ that make the theory works so well. That is what Grothendieck answered by defining *basic localizers*. Indeed, what you need is just a class $W$ of functors between small categories such that: $W$ is weakly saturated (which means it contains identities, it satisfies a two out of three axiom, and if $i$ has a retraction such that $ir$ is in $W$, then $i$ (and thus $r$) is in $W$) ; if $A$ is a small category which has a terminal object, then $A \to e$ is in $W$ ($e$ stands for the point category) ; and $W$ satisfies the relative version of Quillen's Theorem A. That is all you need to develop the theory of test categories. Grothendieck then proceeds to rewrite all the theory with respect to an arbitrary basic localizer replacing $\mathcal{W}_{\infty}$, the classical weak equivalences of $Cat$.Therefore, for *every* basic localizer $W$, there are notions of $W$-weak test category, $W$-local test category, $W$-test category, $W$-strict test category and so on. Truncated homotopy types provide instances of basic localizers $\mathcal{W}_{n}$ for every $n \geq 0$, but there are many others.

And here is a theorem: for every basic localizer $W$, for every $W$-test category $A$, there is a closed model category structure on the category of presheaves on $A$, the weak equivalences of which are those defined above (so that, in particular, the localized category is equivalent to the localized category $W^{-1}Cat$) and the cofibrations of which are the monomorphisms. In fact, you have to make a slight set theoretic assumption for this result to hold (namely, that the basic localizer is *accessible*, that is, it is the smallest one containing some *set* of arrows). It was conjectured by Grothendieck and proved by Cisinski.

OK, now it might still be unclear as to what are the advantages of this theory. One of them is that you can work with other basic localizers than the classical one (the $W_{\infty}$ of above). Classical weak equivalences are related to Artin-Mazur equivalences in slice presheaves toposes, and these can be replaced, for instance, by any other topos morphisms defined by cohomological properties. (See the first paragraph of page 12 of Maltsiniotis's book, for instance.)

There are much more stuff in Grothendieck's homotopy theory, but I shall limit myself to that now.

By the way, there has been a very nice expository talk (in French) by Maltsiniotis on Grothendieck's 1980's work at IHES two years ago:

http://www.dailymotion.com/video/x8jsnw_colloque-grothendieck-georges-malts_tech.

EDIT: I just added some details and thought I could elaborate on two points of Jacob Lurie's answer as well in the language of Grothendieck's homotopy theory (which I of course do not claim to be better). When he states that the (op)-siftedness of the simplicial category guarantees "a nice connection between the homotopy theory of simplicial sets and the homotopy theory of bisimplicial sets", I guess the key result he is alluding to is the classical "bisimplicial lemma", which states that, if $f : X \to Y$ is a bisimplicial morphism such that $f_{n,.} : X_{n,.} \to Y_{n,.}$ is a simplicial weak equivalence for every $n \geq 0$, then $\delta^{\ast}(f):\delta^{\ast}X \to \delta^{\ast}Y$ is a simplicial weak equivalence. Here, $\delta : \Delta \to \Delta \times \Delta$ stands for the diagonal functor, and $\delta^{\ast}$ for the induced functor which send a bisimplicial set $X$ to the simplicial set $n \mapsto X_{n,n}$. I would like to point out that a similar result holds for every *totally aspherical category*, that is, a small category $A$ such that the functor $A \to e$ is a weak equivalence (which means that it belongs to the basic localizer we are considering) and such that (one among many equivalent properties) the diagonal functor $A \to A \times A$ is aspherical (which means that for every $(a_{1}, a_{2}) \in A \times A$, the comma category $\delta \downarrow (a_{1}, a_{2})$ is aspherical). For such a category $A$, whenever $f$ is a morphism in the category of presheaves $\widehat{A \times A}$ such that $f_{a,.}$ is a weak equivalence for all $a \in A$, then $\delta^{\ast}f$ is a weak equivalence (in the category of presheaves, see above). The simplicial category $\Delta$ is $W_{\infty}$-totally aspherical, a (non-trivial) fact from which one can deduce the "bisimplicial lemma". The siftedness has to do with the $W_{0}$-total asphericity, therefore I was puzzled as to how to deduce the "bisimplicial lemma" from it (one needs $W_{\infty}$ as basic localizer). It seems Jacob Lurie is tacitly taking the $(\infty,1)$-categorical viewpoint, which makes the two properties equivalent. (Thanks to Georges Maltsiniotis for poiting that to me.)

As to Dold-Kan correspondence, I asked Maltsiniotis if a similar result holds with other Grothendieck test categories, and the answer is that there is no such result in general, but there is already a conjecture in "Pursuing Stacks" regarding an analogous correspondence for any strict test category.

I am not sure many people wanted to read all that but I thought I would share what I knew since this stuff is not written down in any currently available text.