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.
This can be checked with the following two lines of [sage][2]:
```
sage: C = polytopes.cube().intersection(Polyhedron(ieqs = [[15/8,1,1,1]])) 
....: C.graph().is_inscribable()                                                
False

sage: C.is_simple()                                                             
True
```
It's nice that Rivin's criterion is implemented in [sage][2]...

Here's an image of the graph of the "cube without one corner" 3-polytope, which is non-inscribable and simple:

<img src="https://mo271.github.io/mo/373124/cube_without_corner.svg">

  [1]: https://arxiv.org/abs/1910.05241
  [2]: https://www.sagemath.org/