The replacement phone arrived Wednesday (faster than they said, good). I'd already done a manual backup on top of the automatic one, but migration from one phone to another of the exact same type and OS version is easier: connect them via a cable and wait. Basic data transfer happened within an hour, though it took a few hours for apps to get installed and Chrome was being especially finicky for some reason.
My settings were almost all there; I expected to have to do more manual configuration (including re-laying out the icons where I wanted them). Nope, that was all fine. I had to set up each individual app again, though; sometimes that was just a matter of logging in (for example, Tusky or Authy), but sometimes it required redoing everything (email client for my non-Gmail accounts). Chrome had a weird bug where tabs didn't work (!) but the update ("new version available", it kept saying) would hang; after a few reboots it sorted itself out.
There was a feeling of trepidation as I kept asking myself "are you sure you have everything you need?" before doing the factory reset on the old phone, but I finally did that today. It started doing the flashing-display thing during the reset, so I just left it for a while. The documentation says a factory reset can take an hour, so after a couple hours I power-cycled to see where it was.
I was greeted by the "new phone" setup screen, so that worked.
And then it started flashing again. Ha.
Yes, support person, I was right: that's a hardware problem. After another power-cycle (so I could see what I was doing) I shut it down and boxed it up, and tomorrow I will take it to FedEx.
The replacement they sent me was marked as "refurbished", but they are holding the price of a new phone against my credit card, which feels wrong. It's only a problem if the package doesn't arrive in time (which is why I will hand it to a human at FedEx and get a proper receipt), but it's still sleazy. And yes, if they were to charge the card they would add shipping charges, so it's not to offset that.
I've never had to make a warranty claim on a phone before, so I don't know how my experience with Google compares to what I would have had with other vendors. It's something I should try to find out before I buy my next phone, which I hope will be several years from now.