I was determined to sort this out myself. But here we are.
The z-axis spins, but the plastic “tongue” which is spring-clipped on to the side of the Rigid router slips around the vertical gear. The motor actually “rocks” ever so slightly, and I can hear the plastic click around the z-shaft: https://www.youtube.com/watch?v=rpkt5R6gO3k
I have have tried every conceivable tightness level for the nut on the clamp which secures the router in its housing. In fact, I have used a socket wrench and adjusted the tightness while the z motor spun in order to try every possible value. I have also confirmed that I am able to slide the router in and out by hand (assuming the plastic side-pin is retracted with my other hand).
I discovered, like in this post, defects in the casting of the router’s groove. I took a 1/8" metal file to the groove (in the casing) so that the pin sticking out the side of the aluminum router body could slide more easily. I then applied graphite lubricant to both the aluminum body and the inside of the housing.
At this point, I stood the router up so that I could watch it more closely. You can see the z-axis slippage exacerbated, where the attempt to “raise” is actually causing the router to slip and fall downward (negative z). https://www.youtube.com/watch?v=mp0KGLk_k2M
I have tried applying a small bungee cord to the top of the router to attempt to counteract any shearing forces of gravity.
I’m now out of ideas.