As Robert Bryant observed, something like the solubility of $F$ for $u^{(n)}$ needs to be assumed. Then by the trick (due I believe to D'Alembert) of setting $x=(t,u,\dots,u^{(n-1)})$ the equation can always be rewritten $$ \frac{dx}{ds}=f(x). $$ This is what most geometers would call the "standard ODE", wherein $f$ is a smooth vector field on the manifold where $x$ evolves.
In this setting, the space of (maximal connected) solution curves is indeed always a (not necessarily Hausdorff) manifold. I don't know who first wrote this, but according to this recent obituary a contender seems to be J.-M. Souriau, in his book "Structure des systèmes dynamiques". Translation of the relevant passage:
The manifold of motions of a system
Jean-Marie Souriau has observed that the set of maximal solutions of a differential system on a differentiable manifold possesses itself a natural structure of differentiable manifold, not always separated: thus one can speak of the system's manifold of motions. This very simple property is rarely mentioned in the usual courses on differential calculus.