I examined my router's settings, turned off its firewall, and disabled a few other features, but no dice; I was still handoffless and discontinuous.
A few days later, I tried the iCloud login dance again. I logged every one of my devices out of iCloud, then logged them in again. Suddenly, Handoff was starting to work, but not every time. Safari web pages were showing up on other devices, though not on all of them. Emails were being handed off, but, again, not on all my devices.
The iCloud Drive catch
There's an extra gotcha when using iWork apps. Handoff only works if the files you're working on are saved to iCloud Drive. So if you're not getting those apps to work, choose File > Move To, and move your files to iCloud Drive.
After all this work, however, my MacBook Pro still won't work. I've signed out of iCloud and signed back in again, and it's a compatible model. At this point, I've pretty much given up; these features seem not to be worth jumping through this many hoops to get them to work. I do use them to view web pages in Safari, say, first on my iPad, then on my Mac; or to start replying to an email on my iPhone, then switching to my Mac when I realize I need to attach documents. But other than that, I feel like I've done more than I should for little return.
No good solution
Apple really needs some sort of troubleshooting tool to help users find where things are going wrong — similar to the Connection Doctor in Mail, perhaps. Apple has spent a lot of time marketing these features, so the compay owes its users better support. Apple devices used to "just work," and these features just don't.
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