Note to self: the next time GRUB starts exiting with 1 during postinst (when upgrading the kernel), check /boot/grub/device.map. Apparently the last time this happened, I had to change hd0 to point to sda, and this time I had to change it to point back to hda. Without this change grub-probe would fail, causing update-grub and thus the postinst to fail also.
Also, don’t be fooled by update-grub staying quiet; update-grub && echo success lets you know whether it did actually complete or not.