Here's a counterexample to your title question, that is not a counterexample to BCnrd's claim:

Let $Y =\operatorname{Spec} \mathbb{C}[[t]]$. This scheme has a closed point and an open generic point. Let $Z$ be the scheme formed by gluing two copies of $Y$ by the identity map on generic points. This is a non-separated scheme with an action of a group of order two that fixes the generic point and switches the two closed points. Let $X$ be the stack quotient of $Z$ by this action.

$X$ is not an algebraic space, since the image of the generic point of $Z$ has a nontrivial stabilizer. In particular any geometric point (meaning a map from the spectrum of an algebraically closed field) that factors through the generic point of $Z$ will have nontrivial stabilizer. The topological space $|X|$ (defined in *Champs Algebriques* 5.5) only has one closed point, and the automophism group of any element in its equivalence class is trivial.