To be fair, it is not a trivial fact that a simply connected Riemannian manifold whose sectional curvatures are all nonpositive is CAT(0).

This is because they give the definition in the particular case of discrete groups. The "true" definition" replaces "finite" by "compact". But in general, the topology of the group of isometries is defined in such a way that the action is proper almost by definition. So this is definitely not a problem.