The path will appear to be deselected but go ahead and click on the profile (2D pattern) of the spindle to make Follow Me run. It happens in the video you linked to, too. Depending upon how complex the profile it, it may take a moment.
Tour profiles don’t have a face which would be required. Use the Line tool to trace the top edge of the profile so the faces are created. Then there will be a face for Follow Me to work on.
Due to the small size of the overall objects and the short lengths of some of the profile edges, you’ll wind up with gaps. Scale the model up by a factor of 100 before running Follow Me then scale back down. Or better, use the Dave method.
When you’ve finished, you should have something like this:
Hopefully it doesn’t take too long. You’re almost home with the Follow Me bit. It’s just the small stuff and that’s not difficult if you use the Dave method. Here’s an ancient old thing on drawing chess men that shows the Dave method in action. Maybe it’ll help.
Here’s a moving version of what Dave is saying. I move the circle down on the blue axis so that is not in contact with the profile. This also keeps the face on the spindle the same as removing the face of the circle.