I suspect one of the frequently described problems with Secure Boot and/or the CSM setting in the bios.
I've made some progress!
Based on the screen I saw when booted into desktop bios/gpt machine, I simply blindly - the screen is blank - pressed down-arrow and enter and was able to boot from iso.

However, the installer failed at 'grub-install' phase since due to bios error in my machine, in order to properly install new distro I have to pass
efi_no_storage_paranoia
to the Grub entry.
I've tried to fix the issue by running chroot-helper, but
grub-install /dev/sda
reports 'no space on device'.
So, any idea how to install grub post festum?
I've tried to change grub parameters by adding the above parameter and issuing,
update-grub
which goes well, but the issue is how to install Grub on my HDD's MBR?
So, my conclusion is that the problematic issue is
not due to UEFI settings. The problem seems to be with the Grub menu, which is not visible, and therefore I'm not able to properly select an entry and edit it to pass the required parameter.
Any hint?