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.

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