Thank you for the links, Eric. I asked for an agent to call me, and that gentleman cleared up many misunderstandings. About an hour before the call, I stumbled onto the immediate solution. I was showing the problem to my wife when I noticed the clickable word "Edit" in the upper right corner. That's when it struck me: Why would Apple let me edit my password in Settings if the system is supposed to update such things automatically?
Well, that's where the Apple agent helped me. He explained that my MacBook Pro has Keychain, but the iPhone—smart though it may be—has nothing so robust. If tomorrow I change the password for my Apple ID, I'll then need to go to my iPhone 7 and visit Settings >> Accounts & Passwords >> App & Website Passwords and key in the new password. The same thing applies when I change the password for, say, a credit card online site. I checked the listing on my phone for my credit card's password and, sure enough, the old password displayed. I keyed in the current password, and all's well.
Again, thanks for your help.