Nerves of categories are 2-coskeletal, i.e. right orthogonal to the boundary inclusion $\partial \Delta^n \hookrightarrow \Delta^n$ for all $n > 2$.
On the other hand, the singular set of a topological set $X$ is right orthogonal to $\partial \Delta^n \hookrightarrow \Delta^n$ if and only if $X$ itself is right orthogonal to the geometric realisation of $\partial \Delta^n \hookrightarrow \Delta^n$.
This almost never happens.
The existence of many continuous endomaps of $\left| \Delta^n \right|$ that restrict to the identity on $\left| \partial \Delta^n \right|$ means you can easily manufacture more continuous maps $\left| \Delta^n \right| \to X$ with the same restriction to $\left| \partial \Delta^n \right|$ – unless the map is constant on $\left| \Delta^n \right| \setminus \left| \partial \Delta^n \right|$.
Similarly, the existence of many continuous endomaps of $\left| \Delta^2 \right|$ that restrict to the identity on $\left| \Lambda^2_1 \right|$ makes it difficult for $X$ to be right orthogonal to the geometric realisation of $\Lambda^2_1 \hookrightarrow \Delta^2$, unless every continuous map $\left| \Delta^1 \right| \to X$ is constant.
This implies every continuous map $\left| \Delta^n \right| \to X$ is constant, for all $n \ge 0$.
Therefore:
Proposition. The singular set of $X$ is (isomorphic to) the nerve of a category if and only if, for every $n \ge 0$, every continuous map $\left| \Delta^n \right| \to X$ is constant.
For example, this happens if $X$ is discrete, or more generally, totally disconnected.
The $\infty$-category presented by a model category is also called the underlying $\infty$-category of the model category, and there are several equivalent ways to describe it. The idea is that ''a homotopy theory'' is an $\infty$-category, and that every model category encodes a homotopy theory. Therefore, every model category $\mathcal{M}$ should give rise to an $\infty$-category $\mathcal{M}_\infty$ that ''is'' the homotopy theory which $\mathcal{M}$ encodes, and which as such only retains the homotopical data of the model category, throwing away non-homotopical notions such as cofibrations and fibrations.
To actually construct (a model of) $\mathcal{M}_\infty$, therefore, we only want to remember the weak equivalences of the model category. As a first step, then, we map a model category $\mathcal{M}$ to its underlying relative category (also called ''category with weak equivalences'', but there are several meanings to that term) $(\mathcal{M},\mathcal{W})$, where $\mathcal{W}$ denotes the class of weak equivalences in $\mathcal{M}$. Now, you can for instance proceed in the following two ways:
- Via the hammock localization functor $L^H\colon\mathsf{RelCat}\to\mathsf{sCat}$ we can turn $(\mathcal{M},\mathcal{W})$ into a simplicial category $L^H(\mathcal{M},\mathcal{W})$. We apply the derived homotopy coherent nerve functor $\mathbf{R}N^\mathrm{coh}\colon\mathsf{sCat}\to\mathsf{sSet}_\mathrm{Joyal}$, using the Bergner model structure on $\mathsf{sCat}$. The resulting quasicategory $\mathbf{R}N^\mathrm{coh}L^H(\mathcal{M},\mathcal{W})$ is a particular model for the underlying $\infty$-category of $\mathcal{M}$.
- There is a right Quillen equivalence $N_\xi\colon\mathsf{RelCat}\to\mathsf{CSS}$, where $\mathsf{CSS}$ is the complete Segal space model structure on simplicial spaces (see Barwick--Kan, Relative categories: Another model for the homotopy theory of homotopy theories). (Edit: I made a mistake here with which functor exactly to use, but I fixed it now.) There is a further right Quillen equivalence $\mathrm{ev}_{(-)}(0)\colon\mathsf{CSS}\to\mathsf{sSet}_\mathrm{Joyal}$ (which is evaluation of the sequence of spaces at their 0-simplices). So, the right derived functor $\mathbf{R}(\mathrm{ev}_{(-)}(0)\circ N_\xi)$ maps $(\mathcal{M},\mathcal{W})$ to a quasicategory $\mathbf{R}(\mathrm{ev}_{(-)}(0)\circ N_\xi)(\mathcal{M},\mathcal{W})$, which also models the underlying $\infty$-category of $\mathcal{M}$.
These two constructions yield equivalent $\infty$-categories, and in fact the constructions themselves are naturally equivalent in the sense that their underlying $\infty$-functors $\mathsf{RelCat}_\infty\to(\mathsf{sSet}_\mathrm{Joyal})_\infty$ are naturally equivalent. This follows from Toën's result on automorphisms of $\mathsf{Cat}_\infty$. If $\mathcal{M}$ is a simplicial model category, then a third (equivalent) way to describe $\mathcal{M}_\infty$ is as the homotopy coherent nerve $N^\mathrm{coh}(\mathcal{M}^\circ)$ of the simplicial category $\mathcal{M}^{\circ}$ of fibrant-cofibrant objects in $\mathcal{M}$. This is what Lurie usually works with in Higher Topos Theory.
So, you can now put as definition the following: a model category $\mathcal{M}$ presents an $\infty$-category $\mathcal{C}$ if you supply an equivalence $\mathcal{M}_\infty\simeq\mathcal{C}$ of $\infty$-categories, where $\mathcal{M}_\infty$ may be taken to be any of the equivalent $\infty$-categories above. If $\mathcal{M}$ is a monoidal model category, and $\mathcal{C}$ is a monoidal $\infty$-category, you probably want to add the requirement that the equivalence $\mathcal{M}_\infty\simeq\mathcal{C}$ is a monoidal equivalence.
There is a long list of results that tell you which extra structure on model categories carries over to extra structure on $\infty$-categories presented by the model categories. You can therefore ask, given an $\infty$-category $\mathcal{C}$ with some extra interesting structure, for this structure to also exist in a model category $\mathcal{M}$ that presents $\mathcal{C}$ (assuming such an $\mathcal{M}$ exists), and you can ask for this structure to be preserved by the equivalence $\mathcal{M}_\infty\simeq\mathcal{C}$. This is often (implicitly) done.
We like presentations for multiple reasons, among which that you can sometimes explicitly compute things in $\infty$-categories by looking at a convenient model category that presents them. In some sense, having a model-categorical presentation is similar to working in a basis for a vector space: it is not intrinsic to the underlying object you are interested in, but can be very helpful. However, note that an $\infty$-category automatically satisfies quite strong properties if it can be presented by a model category (such as being complete and cocomplete), so in particular not all $\infty$-categories admit such a presentation.
Best Answer
The mistake lies in combining the first two bullets to conclude that $N(\mathcal{C})$ has no nontrivial higher homotopy groups, because $N(\mathcal{C})$ is generally not a Kan complex*. On the other hand, a Kan fibrant replacement of $N(\mathcal{C})$ will generally not be $2$-coskeletal anymore. The rest of the bullets are all correct (the first two as well, for that matter), but since we have found we don't actually have any restrictions anymore on the higher homotopy groups of $N(\mathcal{C})$, you don't get a contradiction anymore.
*It is a Kan complex precisely if $\mathcal{C}$ is a groupoid. But in that case, $N(\mathcal{C})$ is a disjoint union of the nerves of connected components of $\mathcal{C}$, so let us assume for simplicity that $\mathcal{C}$ is connected. Then it is up to equivalence of categories a group, and hence $N(\mathcal{C})$ is a $K(G,1)$ for some group $G$. This indeed has no higher homotopy groups, so in those cases where our initial category $\mathcal{C}$ satisfies that $N(\mathcal{C})$ happens to be a Kan complex, it does hold that its geometric realization has no higher homotopy groups and we have no contradiction.