By Properties of Stacks, Lemma \ref{stacks-properties-lemma-unique-point-better} there exists a field $k$ and a surjective, flat, locally finitely presented morphism $\Spec(k) \to \mathcal{Z}$. Set $U = \Spec(k)$ and $R = U \times_\mathcal{Z} U$ so we obtain a groupoid in algebraic spaces $(U, R, s, t, c)$, see Algebraic Stacks, Lemma \ref{algebraic-lemma-criterion-map-representable-spaces-fibred-in-groupoids}. Note that by Algebraic Stacks, Remark \ref{algebraic-remark-flat-fp-presentation} we have an equivalence $$ f_{can} : [U/R] \longrightarrow \mathcal{Z} $$ The projections $s, t : R \to U$ are locally of finite presentation. As $\mathcal{Z}$ is DM we see that the stabilizer group algebraic space $$ G = U \times_{U \times U} R = U \times_{U \times U} (U \times_\mathcal{Z} U) = U \times_{\mathcal{Z} \times \mathcal{Z}, \Delta_\mathcal{Z}} \mathcal{Z} $$ is unramified over $U$. In particular $\dim(G) = 0$ and by More on Groupoids in Spaces, Lemma \ref{spaces-more-groupoids-lemma-groupoid-on-field-dimension-equal-stabilizer} we have $\dim(R) = 0$. This implies that $R$ is a scheme, see Spaces over Fields, Lemma \ref{spaces-over-fields-lemma-locally-finite-type-dim-zero}. By Varieties, Lemma \ref{varieties-lemma-algebraic-scheme-dim-0} we see that $R$ (and also $G$) is the disjoint union of spectra of Artinian local rings finite over $k$ via either $s$ or $t$. Let $P = \Spec(A) \subset R$ be the open and closed subscheme whose underlying point is the identity $e$ of the groupoid scheme $(U, R, s, t, c)$. As $s \circ e = t \circ e = \text{id}_{\Spec(k)}$ we see that $A$ is an Artinian local ring whose residue field is identified with $k$ via either $s^\sharp : k \to A$ or $t^\sharp : k \to A$. Note that $s, t : \Spec(A) \to \Spec(k)$ are finite (by the lemma referenced above). Since $G \to \Spec(k)$ is unramified we see that $$ G \cap P = P \times_{U \times U} U = \Spec(A \otimes_{k \otimes k} k) $$ is unramified over $k$. On the other hand $A \otimes_{k \otimes k} k$ is local as a quotient of $A$ and surjects onto $k$. We conclude that $A \otimes_{k \otimes k} k = k$. It follows that $P \to U \times U$ is universally injective (as $P$ has only one point with residue field $k$), unramified (by the computation of the fibre over the unique image point above), and of finite type (because $s, t$ are) hence a monomorphism (see \'Etale Morphisms, Lemma \ref{etale-lemma-universally-injective-unramified}). Thus $s|_P, t|_P : P \to U$ define a finite flat equivalence relation. Thus we may apply Groupoids, Proposition \ref{groupoids-proposition-finite-flat-equivalence} to conclude that $U/P$ exists and is a scheme $\overline{U}$. Moreover, $U \to \overline{U}$ is finite locally free and $P = U \times_{\overline{U}} U$. In fact $\overline{U} = \Spec(k_0)$ where $k_0 \subset k$ is the ring of $R$-invariant functions. As $k$ is a field it follows from the definition Groupoids, Equation (\ref{groupoids-equation-invariants}) that $k_0$ is a field. \medskip\noindent We claim that \begin{equation} \label{equation-etale-covering} \Spec(k_0) = \overline{U} = U/P \to [U/R] = \mathcal{Z} \end{equation} is the desired surjective \'etale morphism. It follows from Properties of Stacks, Lemma \ref{stacks-properties-lemma-flat-cover-by-field} that this morphism is surjective. Thus it suffices to show that (\ref{equation-etale-covering}) is \'etale\footnote{We urge the reader to find his/her own proof of this fact. In fact the argument has a lot in common with the final argument of the proof of Bootstrap, Theorem \ref{bootstrap-theorem-final-bootstrap} hence probably should be isolated into its own lemma somewhere.}. Instead of proving the \'etaleness directly we first apply Bootstrap, Lemma \ref{bootstrap-lemma-divide-subgroupoid} to see that there exists a groupoid scheme $(\overline{U}, \overline{R}, \overline{s}, \overline{t}, \overline{c})$ such that $(U, R, s, t, c)$ is the restriction of $(\overline{U}, \overline{R}, \overline{s}, \overline{t}, \overline{c})$ via the quotient morphism $U \to \overline{U}$. (We verified all the hypothesis of the lemma above except for the assertion that $j : R \to U \times U$ is separated and locally quasi-finite which follows from the fact that $R$ is a separated scheme locally quasi-finite over $k$.) Since $U \to \overline{U}$ is finite locally free we see that $[U/R] \to [\overline{U}/\overline{R}]$ is an equivalence, see Groupoids in Spaces, Lemma \ref{spaces-groupoids-lemma-quotient-stack-restrict-equivalence}. \medskip\noindent Note that $s, t$ are the base changes of the morphisms $\overline{s}, \overline{t}$ by $U \to \overline{U}$. As $\{U \to \overline{U}\}$ is an fppf covering we conclude $\overline{s}, \overline{t}$ are flat, locally of finite presentation, and locally quasi-finite, see Descent, Lemmas \ref{descent-lemma-descending-property-flat}, \ref{descent-lemma-descending-property-locally-finite-presentation}, and \ref{descent-lemma-descending-property-quasi-finite}. Consider the commutative diagram $$ \xymatrix{ U \times_{\overline{U}} U \ar@{=}[r] \ar[rd] & P \ar[r] \ar[d] & R \ar[d] \\ & \overline{U} \ar[r]^{\overline{e}} & \overline{R} } $$ It is a general fact about restrictions that the outer four corners form a cartesian diagram. By the equality we see the inner square is cartesian. Since $P$ is open in $R$ we conclude that $\overline{e}$ is an open immersion by Descent, Lemma \ref{descent-lemma-descending-property-open-immersion}. \medskip\noindent But of course, if $\overline{e}$ is an open immersion and $\overline{s}, \overline{t}$ are flat and locally of finite presentation then the morphisms $\overline{t}, \overline{s}$ are \'etale. For example you can see this by applying More on Groupoids, Lemma \ref{more-groupoids-lemma-sheaf-differentials} which shows that $\Omega_{\overline{R}/\overline{U}} = 0$ implies that $\overline{s}, \overline{t} : \overline{R} \to \overline{U}$ is unramified (see Morphisms, Lemma \ref{morphisms-lemma-unramified-omega-zero}), which in turn implies that $\overline{s}, \overline{t}$ are \'etale (see Morphisms, Lemma \ref{morphisms-lemma-flat-unramified-etale}). Hence $\mathcal{Z} = [\overline{U}/\overline{R}]$ is an \'etale presentation of the algebraic stack $\mathcal{Z}$ and we conclude that $\overline{U} \to \mathcal{Z}$ is \'etale by Properties of Stacks, Lemma \ref{stacks-properties-lemma-check-property-covering}.