The solution I hit upon satisfied everyone. I took the boys to the park, where they climbed up the slide and slid down, over and over. I positioned myself at the top of the slide ladder, and then I could spin maybe three yards of yarn before I had to gather it up on the spindle.
Ever since then I've pottered around with yarn. For years, we had sheep, and I made my poor children card their wool while I read aloud to them and spun the fleece into wool (with a spinning wheel, not a spindle), or knit hats. Hat after hat.
What did people use to card wool with before the stainless steel combs I made my poor children labor with? I'd always thought it was teasels. But I tried using them (had to wear gloves), and nope. The prickles on the seedheads break off and it gets ugly quickly. A bit of research reveals that they were used to raise the nap of already woven cloth, not to card raw wool.
Nevertheless. For me, teasels always evoke those long afternoons reading to my poor laboring children.
There is a whole field of them at the Anacortes ferry landing. Teasels, not children.