How can I boot in single-user mode for terminal with grey screen?
It might be easier to boot in target disk mode to erase a machine that’s not healthy enough to boot to recovery. As you know, you’ll need to assemble an external OS and modify it to prevent graphics drivers to get around your failed GPU / logic board. Target disk mode often works in this case since only the firmware is running - no OS.
Both of these require you have another Mac running in my experience.

https://support.apple.com/guide/mac-help/transfer-files-mac-computers-target-disk-mode-mchlp1443/mac
https://support.apple.com/en-us/HT201255

Option is the startup manager you need if Internet Recovery or built in recovery fail.