Eureka!

I have watched Tammy Rizzo's Ply on the Fly video hundreds of times, and have attempted the technique not quite that many times, but always unsuccessfully.  I'm just not that coordinated, yet.

Then this video appeared on Ravelry:

I am also very low tech, and this is my setup to wind a chained plying ball, the core of which is a very small tennis ball:

In the process, I learned that if you lose the loop off your pinky finger, it's really inconvenient.  Not insurmountable, just incredibly inconvenient.  Picking up the loop again will improve with practice, I'm sure.

The wound skein, with all of the colors together as they should be:

It's still not perfect, and most of the problems are caused by the fact that the singles are drafted too thinly in spots.  There are a few bobbles where the singles broke and had to be spliced back together, but this skein is one step closer to the kind of yarn I want to produce.