Here are some comments about the use of topologies in motivic homotopy theory. This is based on the discussion in Morel-Voevodsky's "A^1-homotopy theory of schemes" p.94-95 (MV below), I only add some background and references. I also comment on the differences between the development of the unstable and stable theories. I am not an expert, so please comment/edit out innacurate statements.
First some model category generalities. Various model structures on categories of simplicial (pre)sheaves on any Grothendieck site $(\mathcal{C},\tau)$ can be constructed. For a survey see:
http://nlab.mathforge.org/nlab/show/model+structure+on+simplicial+presheaves
and the references therein. Many of those are Quillen equivalent and can be thought as different presentations of the $(\infty,1)$-category of $(\infty,1)$ $\tau$-sheaves on $\mathcal{C}$. As usual in model category theory, they are suited for deriving different functors (i.e. some natural functors will be Quillen with respect to some structures but not others).
The resulting model categories are all left proper, simplicial and combinatorial, so by a theorem of Smith (see http://nlab.mathforge.org/nlab/show/Bousfield+localization+of+model+categories#Existence )) left Bousfield localizations at a set of maps exist. In particular if $\mathcal{C}$ is a small category of schemes over a fixed $S$ which is stable by $S$-fiber products and contains $\mathbb{A}^1_S$, one can localize at the set of maps $X\times_S \mathbb{A}^1_S\rightarrow X$.
The Morel-Voevodsky category $\mathcal{H}(S)$ can be obtained by this procedure with $\mathcal{C}=Sm/S$, $\tau=Nis$ and using as starting point the so-called injective model structure on simplicial sheaves. It should be clear from the above that many variants are possible, some giving alternative models for $\mathcal{H}(S)$ (using projective model structures, simplicial presheaves instead of sheaves, or even more exotic choices like cubical presheaves, etc.) and some giving different categories (using another topology: Zariski, étale, using the category of all schemes instead of smooth ones, etc.).
The passage to the stable theory via model categories of spectra is also formal (in the sense that it can be done in great generality and with variants) but rather subtle. See e.g. Riou "Catégorie homotopique stable d'un site suspendu avec intevalle" or Ayoub's thesis.
The question now becomes: in which respects are $\mathcal{H}(S)$ and $\mathcal{SH}(S)$ nicer than the alternatives ? And how much does this depend on the choice of the Nisnevich topology ?
Here are some possible answers:
1) Characterisation of $\tau$-local simplicial (pre)sheaves. For a general site (and in particular for the étale site over a general scheme), this is a complicated condition which can be expressed only in terms of a descent condition for hypercovers (see Dugger-Hollander-Isaksen, "Hypercovers and simplicial presheaves"). For the Zariski and the Nisnevich site, one can show a "Brown-Gersten" property: $\tau$-locality can be rephrased as some squares of simplicial sets associated to $\tau$-distinguished squares being homotopy cartesian. See MV proposition 1.16 for the case of the Nisnevich topology.
Aside: this argument has been abstracted by Voevodsky in "Homotopy theory of simplicial sheaves in completely decomposable topologies" and used to compare $\mathcal{H}(k)$ for $k$ a field admitting resolution of singularities with a similar category defined with $\mathcal{C}=Sch/k$ and $\tau=cdh$. In the setting of triangulated categories of mixed motives, this argument can be pushed in various directions to exploit various forms of resolution of singularities, see Cisinski-Deglise, "Triangulated categories of mixed motives", section 3.3.
The Brown-Gersten property in turn plays an important role in MV. For instance, it implies that the property of being $\tau$-local is stable by filtered colimits. It is also used in MV to construct an explicit A^1-localisation functor and to study the functoriality of $\mathcal{H}(S)$. On the other hand I am not sure if the use of the Brown-Gersten property for all this is unavoidable: at least in the context of $\mathcal{SH}(S)$, there are arguments which do not use this (see Riou's paper mentioned above or Ayoub's thesis) and which consequently give some results for the étale topology. On the other hand, in following works of Morel on $\mathcal{H}(k)$, there are really substantial applications of the Brown-Gersten condition, see "A^1-algebraic topology over a field" Chap. 8 and App. A.
2) Homotopy purity (or localisation). The proof of this major theorem in MV requires a topology at least as fine as the Nisnevich topology and to work with $\mathcal{C}=Sm$. The idea is that to reduce the theorem to the case of the closed immersion $Z\rightarrow A^n_Z$ (where one can write explicit $A^1$-homotopies), one uses the local structure of smooth pairs in the étale topology (cf EGAIV 17.12.2) and the fact that étale morphisms to Henselian local schemes (i.e. points of the Nisnevich site) which have a section on the closed point have a section.
3) Compactness properties. The cohomological dimension of the small Nisnevich site on a noetherian scheme is bounded by the Krull dimension, see MV 1.8 and Thomason-Trobaugh E.6.c. This is very different from the case of the étale topology. This has important consequences for the stable theory. It implies in particular that $\mathcal{SH}(S)$ is compactly generated. See e.g. the last paragraphs of Ayoub's thesis. I do not know if there are analoguous statements to be made in the unstable case.
4) Nisnevich descent for motivic cohomology and algebraic K-theory. The descent properties of algebraic K-theory have been studied long before motivic homotopy theory. The definitive, pre-$A^1$-homotopy result (Nisnevich descent for algebraic K-theory of regular schemes) is in Thomason-Trobaugh, and this is used as an input in the proof of representability of algebraic K-theory in MV p.139. On the other hand, algebraic K-theory with integral coefficients does not satisfy étale descent and hence cannot be represented in the étale $A^1$-homotopy category.
It's worth noting first that smooth schemes are essentially the smallest possible category from which one can define the motivic homotopy category: to make sense of $\mathbb A^1$-homotopy and of the Nisnevich topology you need étale extensions of affine spaces in your category, and every smooth $S$-scheme is (Zariski) locally such.
That said, I can think of two fundamental places in the theory where smoothness is crucial, both of which also showcase the relevance of the Nisnevich topology and of $\mathbb A^1$-homotopy.
(1) The first is the localization property already addressed in Adeel's answer, which is itself crucial for many things, such as the proper base change theorem.
A characterizing property of henselian local schemes $S$ (which are the points of the Nisnevich topology) is that for $f:X\to S$ étale every section of $f$ over the closed point $S_0\subset S$ lifts uniquely to a section of $f$ over $S$ ("Hensel's lemma"); this is what makes localization work for sheaves on the small Nisnevich or étale sites. If $f: X \to S$ is smooth, it is still the case that every section $s_0:S_0 \hookrightarrow X$ lifts to $S$ (this uses smoothness in an essential way), but not uniquely. However, once a lift $s:S\hookrightarrow X$ has been chosen, then $X$ can be presented as an étale neighborhood of $S$ in the normal bundle of $s$, and it follows that the Nisnevich sheafification of the "space of lifts" of $s_0$ to $S$ is $\mathbb A^1$-contractible. Thus, in some precise sense, lifts are still unique up to $\mathbb A^1$-homotopy. This is the proof of the Morel-Voevodsky localization theorem in a nutshell.
Another nice consequence of the localization property is that fields form a "conservative family of points" in motivic homotopy theory (at least for the $S^1$-stable theory, though one can also say something unstably), something that could not be achieved using a Grothendieck topology alone.
(2) The second is Cousin/Gersten complexes. This is now specific to the case of a base field $k$, in which localization is useless. The key input here is a geometric presentation lemma of Gabber, a statement of which can be found as Lemma 15 in the introduction to Morel's book (http://www.mathematik.uni-muenchen.de/~morel/Prepublications/A1TopologyLNM.pdf). A consequence of this lemma is that every $\mathbb A^1$-invariant Nisnevich sheaf $F$ (of spaces or spectra, say) is "effaceable" on smooth $k$-schemes, in the sense of Colliot-Thélène-Hoobler-Kahn (https://webusers.imj-prg.fr/~bruno.kahn/preprints/bo.dvi). This implies that the coniveau spectral sequence degenerates at $E_2$ and hence that the Cousin complex
$$ 0 \to \pi_nF(X) \to \bigoplus_{x\in X^{(0)}} \pi_{n}F_x(X_x) \to \bigoplus_{x\in X^{(1)}} \pi_{n-1}F_x(X_x) \to \dots $$
is exact when $X$ is smooth local. Here, $X^{(n)}$ is the set of points of codimension $n$, $X_x=\operatorname{Spec}(\mathcal O_{X,x})$, and $F_x(X_x)$ is the homotopy fiber of the restriction map $F(X_x) \to F(X_x-x)$.
These Cousin complexes are the basis for many computations in motivic homotopy theory, for example in the above-mentioned work of Morel. They can perhaps be viewed as a replacement for cell decompositions in topology.
So, in summary, the smooth/Nisnevich/$\mathbb A^1$ combo simultaneously allows us to (1) reduce questions over general base schemes to the case of base fields via localization, and (2) perform interesting computations in the latter case.
Best Answer
(Don't be afraid about the word "$\infty$-category" here: they're just a convenient framework to do homotopy theory in).
I'm going to try with a very naive answer, although I'm not sure I understand your question exactly.
The (un)stable motivic ($\infty$-)category has a universal property. To be precise the following statements are true
(see Dugger Universal Homotopy Theories, section 8)
(see Robalo K-Theory and the bridge to noncommutative motives, corollary 2.39)
These two theorems are saying that any two functors that "behave like a homology theory on smooth $S$-schemes" will factor uniquely through the (un)stable motivic ($\infty$-)category. Examples are $l$-adic étale cohomology, algebraic K-theory (if $S$ is regular Noetherian), motivic cohomology (as given by Bloch's higher Chow groups)... Conversely, the canonical functor from $\mathrm{Sm}_S$ to the (un)stable motivic ($\infty$-)category has these properties. So the (un)stable motivic homotopy theory is in this precise sense the home of the universal homology theory. In particular all the properties we can prove for $\mathcal{H}(S)$ or $SH(S)$ reflect on every homology theory satisfying the above properties (purity being the obvious example).
Let me say a couple of words about the two aspects that "worry you"
$\mathbb{A}^1$-invariance needs to be imposed. That's not surprising: we do need to do that also for topological spaces, when we quotient the maps by homotopy equivalence (or, more precisely, we need to replace the set of maps by a space of maps, where paths are given by homotopies: this more complicated procedure is also responsible for the usage of simplicial presheaves rather than just ordinary presheaves)
Having more kinds of spheres is actually quite common in homotopy theory. A good test case for this is $C_2$-equivariant homotopy theory. See for example this answer of mine for a more detailed exploration of the analogy.
Surprisingly, possibly the most problematic of the three defining properties of $SH(S)$ is the $\mathbb{A}^1$-invariance. In fact there are several "homology theories" we'd like to study that do not satisfy it (e.g. crystalline cohomology). I know some people are trying to find a replacement for $SH(S)$ where these theories might live. As far as I know there are some definitions of such replacements but I don't think they have been shown to have properties comparable to the very interesting structure you can find on $SH(S)$, so I don't know whether this will bear fruit or not in the future.