Thanks to everybody who responded. I did eventually figure this out. Curves and lines can be divided... shapes cannot—except by Boolean operations.
Like Steve1, I am a refugee from Freehand, the finest, fastest and most reliable and intuitive vector program ever built. Steve, I'm here to tell you I've tried 6 different alternatives to replace Freehand, and Affinity Designer comes the closest. It still has puzzling omissions, along with all its delightful additions. But it's stable, reliable and consistent. Not to mention astonishingly affordable. I sincerely hope that the very moderate cost does NOT mean the company will eventually be hurting for cash and have to sell out, as did Freehand. I have 25 years of Freehand files that are now, sadly, only accessible with the great pain of using an old Mac and sending them to my current machine. I hope the AD does not go the same route.
But Steve1, in the mean time, I have to admit that AD is a great, modern vector program that does not insist on a cloud model. Hallelujah! (Now we'll both hope to see a knife tool sometime in the future... and after that, I'll hope for named colors!)