This is a fascinating problem that ultimately involves the ability (or not) to connect to Apple's servers. I think the steps to resolve it are:
1. Reset network settings
2. Perform a soft reset on the phone by holding down the "Home" and wake/sleep buttons until the phone restarts, the Apple symbol appears and then disappears
3. Another thing one might try is to go into Settings > iCloud and see whether, in the second white field, you see a spinning cursor instead of "Set Up Family Sharing…" or "Family Sharing" in blue. Also, see whether you're able to access your iCloud account info when you click on the first white field with your name and AppleID. When I was having the download/update problem, I could not access my iCloud account info and always saw the spinning cursor in the second white field.
I also tried a restore (DFU restore), setting the iPhone up as a new phone, but the problem still recurred, so I had my iPhone replaced. Then I set up the new iPhone using my backup from the old, and the problem recurred (another symptom is that when you go into the App Store > Updates > Purchased, you'll see that the "Not on this iPhone" tab will *not* properly recognize apps that are *already* on your phone). A soft reset in this case (after hardware replacement) seems to have resolved the issue.