Here's an additional answer just in case you might want to create an outline object in Cycles. Of course you can use something like Freestyle or a Line Art modifier, this is just an alternative method. Although I think you're better off with Eevee since you use a lot of Shader to RGB nodes which simply cannot be used in Cycles.
As I said in the comments, Cycles does not use Backface Culling, it is only available in Eevee. Nevertheless there is the Geometry node in the Shader Editor which has an Backfacing output. This can be used to mix between a shader that creates the outline and a fully white Transparent BSDF to make the backfacing parts invisible.
I also woudn't recommend using a Principled BSDF for the outline because I guess you want a solid color outline and not reflections or anything on it. For a solid black outline the shader is very simple to create: Take a Mix Shader and plug a Transparent BSDF in the second Shader input, then plug the Backfacing output of a Geometry node in the Fac. The backfaces will be transparent and the front faces will be pure black without any shading (since there is nothing in the first Shader input):
This is when using your model, where you have already flipped the faces inside out to make the Backface Culling work. However, with the Backfacing output you wouldn't even have to flip the normals - you would simply plug the Transparent BSDF in the first Shader input.
Now this was the setup for a simple shadeless black outline. If you want to use a color (that's also shadeless), you have to plug in an Emission shader with a Strength of 1 into the other input. But that brings up a problem - the emissive color now illuminates the inner object from all sides becauses it encloses it completely:
So in order to make the outline only visible for the camera and not affect the rest of the scene, you can go to the Object Properties > Visibility > Ray Visibility and disable everything except Camera:
Now you just have the outline without any light shining on the other objects.
What is a little bit inconvenient though is that would have to change the Visibility settings on each object that you want to work as an outline. With a slightly more complex shader setup this won't be necessary though: if you take another Mix Shader node and a Light Path node, you can use the Is Camera Ray as mix factor between the Transparent BSDF plugged in the first Shader input_ and the result of the first Mix Shader into the second input. This makes the camera see the outline correctly while for the rest of the scene objects the outline is completely invisible and thus does not illuminate anything. This way you can simply put this material on any object that should work as an outline without the need to change the Ray Visibility settings (although for Eevee you still would have to change settings per object).
In the above screenshot the first Mix Shader is set up for non-flipped normals, just in case you're wondering. By the way, with this setup you can use the outline in both Cycles and Eevee. To make it work in both you can use the following variations:
- The normals of the outline objects are flipped - in this case, the Transparent BSDF goes into the second input and for Eevee you can either enable Backface Culling or set Blend Mode to something else then Opaque like Alpha Hashed for example.
- The normals are oriented correctly - then you plug the Transparent BSDF in the first input and you cannot use Backface Culling in Eevee but have to set Blend Mode to something like Alpha Hashed.
Now the only problem you might have is when you have Bloom activated in Eevee or want to add some glow in the Compositor, the Emission shader will be glowing. You might have to adjust the Threshold to some value that the outline does not start to glow and other objects that should glow are brighter, or work with masking in the Compositor (but that's stuff for a different question on how to separate those things in compositing).