Let $X = [P/G]$ be a smooth finite type separated DM-stack over $\mathbb C$ given as the quotient of a smooth projective scheme $P$ by the action of a smooth (finite type separated) reductive group scheme $G$.

At atlas of $X$ is an etale morphism $U\to X$ with $U$ an algebraic space.

I'm interested in the converse of the following statement:

Thm. If the coarse moduli space of $X$ is a scheme (i.e., not just an algebraic space) then every atlas $U$ of $X$ is a scheme.

Proof. The map from $U$ to the coarse moduli space $X_c$ of $X$ is quasi-finite separated. Therefore $U$ is a scheme by Knudson Cor. II.6.16., p. 138 QED

So the converse would read as follows:

Q. Suppose that every atlas $U$ of $X$ is a scheme. Is the coarse moduli space of $X$ a scheme?

I expect the answer to be negative, but can't find a good example. One possibility would be to start with a simply connected (non-representable) algebraic space of dimension two with only one stacky point. Is every atlas of such an algebraic space a scheme?