I don’t know what is going on with your PCB, but KiCad’s 3D viewer is definitely able to show the internal copper layers.
Here is an example, taken from the Olinuxino A64 Rev C (Also a 6 layer PCB):
And the settings used for the 3D viewer (Preferences / 3D viewer)
Some noteworthy notes:
- Turn off Solder mask layers (because they cover most of the outside of the PCB).
- Turn off Board Body (which also obfuscates the internals)
- Turn off Realistic Mode This gives each copper layer a different color (I think the same as in the PCB editor).
- Experiment a bit with the settings. Different sorts of inspections work best with different settings.