This seems to be about the closest thing that is possible.

If you try to force the hex faces to be planar like you did, that will result in non-planarity in the pentagon faces.

If you try to force the pentagon faces to be planar like I did, that results in non-planarity of the hexagon faces around the sides.

The angles don't work out for both to be planar at the same time - only certain arrangements of polygons can be made into stuff like this.

- Michael