Yes. In fact you don’t need the hypothesis “every atlas is a scheme”; the existence of one étale atlas by a scheme already forces the coarse moduli space to be a scheme.

Here is a convenient way to see it.

• Let π: X → Y be the coarse moduli space of a separated DM stack X with finite inertia (so Y is an algebraic space).

• Pick an étale atlas p: U → X with U a scheme (such an atlas always exists for a DM stack).

• By the basic properties of coarse moduli spaces (Keel–Mori), the composite f: U → X → Y is locally quasi-finite and separated. One way to check this is étale locally on Y: étale locally on Y the stack X is a quotient [V/G] by a finite group G, and then U ×_Y (–) → (–) factors through V → V/G which is finite; separatedness and local quasi-finiteness descend from this local model.

• Apply Zariski’s Main Theorem for algebraic spaces (Stacks Project, Tag 04S7): there exists a factorization U ↪ U' → Y where U → U' is an open immersion and U' → Y is finite. Since U → Y is surjective, the finite morphism U' → Y is also surjective.

• Being finite is an affine property, and the property “is a scheme” is fpqc local on the base for affine morphisms (Stacks Project, Tag 03GN). Choose an étale (hence fpqc) cover Y0 → Y with Y0 a scheme. Then U' ×_Y Y0 is finite over Y0 and hence a scheme; by fpqc descent for affine morphisms, U' itself is a scheme.

• Finally, there is a finite surjective morphism U' → Y from a scheme to the algebraic space Y. This forces Y to be a scheme (Stacks Project, Tag 03I1).

This proves that the coarse moduli space Y is a scheme. As noted, the argument only uses the existence of one scheme atlas U → X; the stronger hypothesis that every atlas is a scheme is not needed.

Remarks:

• The above local model [V/G] with G finite is standard (e.g. Keel–Mori, or Stacks Project, Tag 06N8), and is the key input ensuring f: U → Y is locally quasi-finite and separated.
• An alternative proof is to write R = U ×_X U (a scheme, since X is DM and U is a scheme). The map R → U ×_Y U is finite (again by the local [V/G] model). Thus R ⇒ U is a finite equivalence relation. Finite equivalence relations on schemes are effective in the category of schemes, so the quotient U/R exists as a scheme, and it identifies with Y by the universal property of the coarse moduli space.