Let us prove this in case $\mathcal{P} = (\beta)$. Let $y \in |Y|$ be a point. We have to show that $y$ can be represented by a monomorphism from a field. Choose a point $x \in |X|$ with $f(x) = y$. By assumption we may represent $x$ by a monomorphism $\Spec(k) \to X$, with $k$ a field. By Lemma \ref{lemma-R-finite-above-x} it suffices to show that the projections $\Spec(k) \times_Y \Spec(k) \to \Spec(k)$ are \'etale and quasi-compact. We can factor the first projection as $$ \Spec(k) \times_Y \Spec(k) \longrightarrow \Spec(k) \times_Y X \longrightarrow \Spec(k) $$ The first morphism is a monomorphism, and the second is \'etale and quasi-compact. By Properties of Spaces, Lemma \ref{spaces-properties-lemma-etale-over-field-scheme} we see that $\Spec(k) \times_Y X$ is a scheme. Hence it is a finite disjoint union of spectra of finite separable field extensions of $k$. By Schemes, Lemma \ref{schemes-lemma-mono-towards-spec-field} we see that the first arrow identifies $\Spec(k) \times_Y \Spec(k)$ with a finite disjoint union of spectra of finite separable field extensions of $k$. Hence the projection morphism is \'etale and quasi-compact. \medskip\noindent Let us prove this in case $\mathcal{P} = decent$. We have already seen in the first paragraph of the proof that this implies that every $y \in |Y|$ can be represented by a monomorphism $y : \Spec(k) \to Y$. Pick such a $y$. Pick an affine scheme $U$ and an \'etale morphism $U \to X$ such that the image of $|U| \to |Y|$ contains $y$. By Lemma \ref{lemma-UR-finite-above-x} it suffices to show that $U_y$ is a finite scheme over $k$. The fibre product $X_y = \Spec(k) \times_Y X$ is a quasi-compact \'etale algebraic space over $k$. Hence by Properties of Spaces, Lemma \ref{spaces-properties-lemma-etale-over-field-scheme} it is a scheme. So it is a finite disjoint union of spectra of finite separable extensions of $k$. Say $X_y = \{x_1, \ldots, x_n\}$ so $x_i$ is given by  $x_i : \Spec(k_i) \to X$ with $[k_i : k] < \infty$. By assumption $X$ is decent, so the schemes $U_{x_i} = \Spec(k_i) \times_X U$ are finite over $k_i$. Finally, we note that $U_y = \coprod U_{x_i}$ as a scheme and we conclude that $U_y$ is finite over $k$ as desired. \medskip\noindent Let us prove this in case $\mathcal{P} = reasonable$. Pick an affine scheme $V$ and an \'etale morphism $V \to Y$. We have the show the fibres of $V \to Y$ are universally bounded. The algebraic space $V \times_Y X$ is quasi-compact. Thus we can find an affine scheme $W$ and a surjective \'etale morphism $W \to V \times_Y X$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-affine-cover}. Here is a picture (solid diagram) $$ \xymatrix{ W \ar[r]  \ar[rd] & V \times_Y X \ar[r] \ar[d] & X \ar[d]_f & \Spec(k) \ar@{..>}[l]^x \ar@{..>}[ld]^y \\  & V \ar[r] & Y } $$ The morphism $W \to X$ is universally bounded by our assumption that the space $X$ is reasonable. Let $n$ be an integer bounding the degrees of the fibres of $W \to X$. We claim that the same integer works for bounding the fibres of $V \to Y$. Namely, suppose $y \in |Y|$ is a point. Then there exists a $x \in |X|$ with $f(x) = y$ (see above). This means we can find a field $k$ and morphisms $x, y$ given as dotted arrows in the diagram above. In particular we get a surjective \'etale morphism $$ \Spec(k) \times_{x, X} W \to \Spec(k) \times_{x, X} (V \times_Y X) = \Spec(k) \times_{y, Y} V $$ which shows that the degree of $\Spec(k) \times_{y, Y} V$ over $k$ is less than or equal to the degree of $\Spec(k) \times_{x, X} W$ over $k$, i.e., $\leq n$, and we win. (This last part of the argument is the same as the argument in the proof of Lemma \ref{lemma-descent-universally-bounded}. Unfortunately that lemma is not general enough because it only applies to representable morphisms.)