The easies fix is to connect your other system to your broken system via FireWire or Thunderbolt to gain access to the GPU driver Kext and just delete it. This was the solution with older OS-X releases now Apple has SIP which gets in the way. And deleting the Kext won’t allow your system to sleep properly with the newer OS’s
I haven’t had a 2011 with a bad GPU pass my desk for quite awhile so I haven’t tested things with High Sierra or newer.
Here’s a good write up on whats involved [https://www.jeffgeerling.com/blog/2017/fixing-2011-macbook-pro-booting-grey-screen-amd-radeon-video-glitch|Fixing a 2011 MacBook Pro booting to a Grey Screen - AMD Radeon Video Glitch]