I'm having a hard time understanding the axial motion from the coarse adjustment. From [0]: The Ball Screw in Figure 9 has a 21 mm range. As the ball screw nut moves up, the housing moves up too -- but then the two halves of the tumbler coupling get displaced in such a way that doesn't seem to support a 21mm displacement. How does the tumbler coupling stay mated? (I feel like I'm missing something obvious?)

Why are you supposing that the two halves of the coupler are moving apart? They are both attached to the moving section

Figure 9 implies that the coupler drives the blue gear (shaft of lead screw).

But let's assume you are right: Then how does the blue gear at top of the lead screw simultaneously turn the shaft and move upwards too?

I believe that everything in figure nine moves together vertically except the yellow/orange ball screw nut and pink horizontal thingie, these are fixed to the satellite.

The sensor and stabilizer on the right side are flexible.