I have tried to recreate the part with the 2 "discs", their mounting and the red bar with the purple box in the middle. However, I would like to achieve the same effect of realism as on the left hand side, where there's added depth to the objects. But I am not sure how to achieve this in Illustrator. Thanks for any tips!
Try to present them as extrudes (see the 3D effects for objects). Give a different extrusion height for those parts that in reality have different height. You must combine your model from several pieces to be able to specify the extrusions independently. The shadows and reflections of the light makes the difference.
Without 3D effects, the light and shadows must be expressed by inserting them manually. We have seen in old atworks, how well shadows, glows and shines can be painted. But that job needs remarkable talent and practice. At least me as well as other ordinaries must use 3D effects to get something worthy enough to be shown.
You have tried to insert a radial gradient onto the grey parts. Unfortunately thar's considered usually to be a ball or other convex surface. The highlights should be mostly at the edges on the planar surfaces. Easy to say - try 3D, like me!
But if you have some talent, you probably soon learn, how at least acceptably convincing lights & shadows are manually doable. Start by trying to repeat what 3D extrusion produces (after understanding and adjusting all its parameters)