By default, grub will WANT or TRY to use the UUID as the root device, UNLESS in /etc/default/grub you enable the feature of GRUB_DISABLE_LINUX_UUID=true.
Grub uses scripts in is /etc/grub.d which are called when you run update grub.
NOTE: It does not matter if your fstab is updated to use UUID, this script does not care about fstab or the current root filesystem.
NOTE: /dev/disk/by-uuid is really just a series of UUIDs in that directory that are symlinked to their actual device name.
NOTE: Do not modify grub.cfg manually, as the next kernel update will cause grub to revert back to the device name.