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

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:

```
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...

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

I just checked that this is the *smallest* non-inscribable simple 3-polytope: all other simple 3-polytopes with up to 10 vertices are inscribable.

"Six Topics on Inscribable Polytopes"by Padrol and Ziegler (p. 409 in"Advances in Discrete Differential Geometry"). $\endgroup$ – M. Winter Oct 2 '20 at 10:46