Not all simple polytopes are incribable, e.g. the dual of the cyclic polytope $C_4(8)$ is simple and not inscribable, as shown recently in [*Combinatorial Inscribability Obstructions for Higher-Dimensional Polytopes* by Doolittle, Labbé, Lange, Sinn, Spreer and  Ziegler][1]


In dimension $3$, there is a combinatorial criterion by Rivin describing inscribabilty completely. I think already a cube with corner cut, which is simple, will be a non-inscribable $3$-polytope.

  [1]: https://arxiv.org/abs/1910.05241