Let me assume that $S$ is a regular Noetherian scheme (for example a field). Then algebraic K-theory is a motivic spectrum, and in fact it is represented by the $\mathbb{P}^1$-spectrum that is $BGL_\infty\times\mathbb{Z}$ in each level (so it is a $\mathbb{P}^1$-periodic motivic spectrum).
This is theorem 4.3.13 in
Morel, Fabien; Voevodsky, Vladimir, $\bf A^1$-homotopy theory of schemes, Publ. Math., Inst. Hautes Étud. Sci. 90, 45-143 (1999). ZBL0983.14007.
A correct proof of the above result can be found in this survey (thanks to Marc Hoyois for pointing out that the original proof was incorrect).
The proof of the fact that algebraic K-theory is a motivic spectrum goes exactly how you described:
- It satisfies Nisnevich descent on qcqs schemes by Thomason-Trobaugh's main theorem.
- It is $\mathbb{A}^1$-invariant on regular Noetherian schemes (which every scheme smooth over $S$ is ) by Quillen's fundamental theorem of K-theory
- It is $\mathbb{P}^1$-periodic by the projective bundle formula (also in Thomason-Trobaugh)
When $S$ is not regular noetherian the situation is more complicated. You can still represent $K$ by some version of the infinite Grassmannian (proposition 4.3.14 in Morel-Voevodsky), but this object won't be $\mathbb{A}^1$-invariant anymore. What you can consider is the homotopy K-theory presheaf $KH=Sing_* K$. This object is indeed represented by $BGL_\infty\times \mathbb{Z}$. This result has been announced in Voevodsky's ICM address and proven in
Cisinski, Denis-Charles, Descent by blow-ups for homotopy invariant $K$-theory, Ann. Math. (2) 177, No. 2, 425-448 (2013). ZBL1264.19003.
(thanks to Marc Hoyois for the reference to this result)
(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
Theorem: Every functor $E:\mathrm{Sm}_S\to C$ to a(n $\infty$-)category $C$ that
- is $\mathbb{A}^1$-invariant (i.e. for which $E(X\times \mathbb{A}^1)\to E(X)$ is an equivalence)
- satisfies "Mayer-Vietoris for the Nisnevich topology" (i.e. sends elementary Nisnevich squares to pushout squares)
factors uniquely through the unstable motivic ($\infty$-)category.
(see Dugger Universal Homotopy Theories, section 8)
Theorem: Every symmetric monoidal functor $E:\mathrm{Sm}_S\to C$ to a pointed presentable symmetric monoidal ($\infty$-)category that
- is $\mathbb{A}^1$-invariant
- satisfies "Mayer-Vietoris for the Nisnevich topology"
- sends the "Tate motive" (i.e. the summand of $E(\mathbb{P}^1)$ obtained by splitting off the summand corresponding to $E(\mathrm{Spec}S)\to E(\mathbb{P}^1)$) to an invertible object
factors uniquely through the stable motivic ($\infty$-)category.
(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.
Best Answer
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.