Make a shape with holes - draw the outline of the footprint and subtract from it with pathfinder panel function "Minus front" grouped rectangles. As well you can create the holes with Object > Compound path > make.
Apply 3D effect extrude to that shape. Make a hole to the background with zero deep copy of the extrusion. An example
A grey footprint outline, cyan rectangles on it, the rectangles are grouped
Select all, goto Object > Compound Path > Make. Now you have the holes.
Make the 3D extrusion
Copy the extruded object. Via the appearance panel reopen the 3D effect dialog of the copy and set the extrusion depth =0. Expand the appearance, ungroup release the compound path and delete the hole shapes to have the outline of the footprint in the right viewing angle. Save the dialog settings somewhere, because you can need them later.
5...7 Make a hole to the background with the outline and place the extrusion. You can also use image 4 as a clipping mask for the extrusion (3) so there's no need to make more holes.
If also the footprint outline needs some depth, you can make another extrusion with the background with the hole. Use the original footprint outline, copy the viewing settings from the original extrusion dialog
About comment " I was on the right track but moving in the opposite direction"
The track actually is a loop, so the direction was right. See this:
The grey pattern in the left is like yours and the orange one is its outline. In the middle the grey shape is extruded. The watching direction is "towards the ceiling".
The orange shape has got the same extrusion, but the depth is zero. In the right the orange extrusion has been used as a clipping mask. The direction of the light and masking together turn extrusion to intrusion.
The opposite is as possible and very easily an image of a carving looks like it's an extrusion. That can really be a problem when one tries to make a drawing.