Why not use the 3D Extrude & Bevel effect?
This will do must of the work for you, at least the central ‘wheel’.
I though I would just place a guide every 36 degrees (there are 10 spokes) and then drop in some rectangles and use the white arrow tool to distort their tops and bottoms.
This all indicates that you need to polish up your knowledge of perspective drawing (or are entirely innocent of it).
36° is only o.k. if you are looking at the thing from the top.
I though I would just place a guide every 36 degrees...
You're basically on the right track.
The drawing you're trying to emulate is a parallel perspective, not a converging (vanishing point, conic, etc.) perspective.
Drawing the radii separated by 36° is correct when looking at the diagram "in the flat" as if from above the vertical axle. But you want to draw it in parallel perspective.
The construction for that is a simple matter of drawing it "in the flat" (thinking of that as the diagram's "top view") and then scaling it vertically (thinking of that as "tilting" the top view relative to your line of sight).
In a diagram like this, the amount you scale the top view vertically is arbitrary; just scale it as much as you like. If you were actually rendering a physical 3D object, you would want the scaling of the top view to be in correct proportion to the object's foreshortened height.
In other words, if the rectangles along the spokes were actually supposed to be squares, you'd want the vertical "extrusion" distance to be in correct foreshortened proportion to the foreshortening of the circles. That's not really an issue here. I just mention it because whenever one gets into this kind of discussion someone erroneously assumes the extrusion distance in parallel perspective is arbitrary or just "eyeballed"; but it is usually not.
The means you use by which to remove the hidden lines and color-fill the remains is up to you, and is dependent upon the particular drawing program you're using and its version.
In Illustrator, depending on version capabilities and personal preference, some would individually cut / join paths, others would advise you to use Pathfinder Divide (and would fail to mention that it removes path portions outboard enclosed areas), others would suggest Pathfinder Outline (and would fail to mention that it does not generate closed paths and that it removes all strokes and fills), others with later versions would sing the praises of LivePaint (and fail to mention the need to Expand and then Pathfinder Merge the results), others with still later versions would glorify the Shape Builder tool (and fail to mention the keyboard shortcut for deleting open-ended portions of paths).
So it's kind of up to you to read and work through the documentation of how to work the program. My purpose here is just to explain the foregoing principle which works in pretty much any drawing program.
Despite 30 years of their existence, mainstream 2D drawing programs are still very piecemeal in their provisions for 2D parallel perspective drawing. (Illustrator only recently acquired a more than basic provision for ordinary vanishing point perspective--and directly copied that from FreeHand). And what little provisions you can find, none are really complete. It's a sadly-neglected and commonly-needed aspect of commerical illustration.