I don't see anyone else reporting problems with this update so there may be something unusual with my hardware or filesystem, but I thought I should report what happened with this server in case it is helpful for someone.
I downloaded an ISO of CentOS 6.9 minimal, burned a CD-ROM and rebooted the server. I used rescue mode, and after looking around at the filesystem for a while I was reasonably confident the data was intact, so I re-installed GRUB:
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chroot /mnt/sysimage
/sbin/grub-install /dev/md0
My boot disk uses software RAID1, which is what the /dev/md0 is about. Two drives /dev/sda and /dev/sdb, raided and partitioned for /, /sboot, /tmp and swap. There's a couple other drives in the system as well, but they only hold data and aren't part of this RAID array.