Restore/copy GPT+EFI disk backups to new disk and make bootable

I've been going in circles. My situation is as follows:

• old disk died, it was a GPT disk booting EFI with 3 partitions, sda1=fat32 (flagged esp,boot and containing /boot and /EFI), sda2=etx4 (containing the rest, as well as the typical /boot), sda3=swap (linuxswap);
• I have backups (made using rsync -aAXH) of the contents of both sda1 and sda2
• I have a new disk and a bootable Ubuntu 20.04.3 USB stick

I want to restore the backups onto the new disk and make it bootable.

What I've tried (unsuccessfully):

• boot from the usb stick, gparted sda and made it GPT, then recreated the same 3 partitions, sda1=fat32 and flagged it esp,boot, sda2=ext4, sda3=swap
• restored all contents from old sda1 backup to new sda1, and from old sda2 to new sda2
• edited (sda2)/etc/fstab and updated the UUIDs with the new values from lsblk
• edited (sda1)/EFI/ubuntu/grub.cfg and updated the UUID of the search.fs_uuid line
• installed grub-efi-amd64 and attempted grub-install /dev/sda

The last line fails with:

Installing for x86_64-efi platform
grub-install: error: cannot find EFI directory

Obviously, ignoring that and rebooting does show the grub menu, but doesn't actually boot once I choose an entry. I think the only step that's missing is to update the UUIDs in (sda2)/boot/grub/grub.cfg since the ones there are still the old ones. This is normally done with update-grub if I'm not wrong (?).

Can someone please help with a list of steps and commands? What am I missing?

FULL DISCLOSURE: I'm attempting this first in a VMware virtual machine, whose .vmx was modified to use UEFI firmware (add firmware = "efi" in the .vmx). This fully emulates a UEFI machine as far as I'm aware.

Answering my own question, but I'm not accepting it as answer yet, because I'm still interested in a clear set of steps to fix it manually.

Using the pointers by @oldfred in a comment above, I used boot-repair which fixed it in one go.

I installed boot-repair after booting from a live CD, using the ppa (and not the ISO, which apparently is buggy), i.e.

sudo apt-add-repository ppa:yannubuntu/boot-repair
sudo apt-get update
sudo apt-get install boot-repair
sudo boot-repair

then chose to apply all default fixes, which worked in my case (note that I had backups and was in a VM so I could afford to be adventurous).

In the end it seems that it did was what I suspected: updated the UUIDs in grub.cfg, which would have been a bit of a pain to do manually, so I was indeed close.

You need like:

pacstrap /mnt base linux linux-firmware
pacstrap /mnt grub-bios

when startup PC also configure bios to Bios in secure boot (not EFI or how there...)