Sorry to read having so many issues.
From the series of events, in you reply, it would appear the Big Sur has not installed but is occupying a lot of space. Thus, the restarting and shutting down before it fully loads.
There might be a way to clear off some files by using Target Disk Mode
That, or on the older 2011 computer - make a a bootable installer of High Sierra ( yes going backwards ) and use that to Wipe the iMac back to High Sierra and commence thereafter the upgrade to Big Sur.
The below method will Wipe ALL DATA on the iMac including the Recovery Volume.
To perform this action will require booting from a Bootable Installer.
This will have to be performed from a Qualifying Computer to run the version of macOS to be made on the Bootable Installer. Example : Bootable Installer of High Sierra would have to be done on a computer that Qualifies to run High Sierra.
Once that is done read on for preparing the Destination computer >>
Only works on Intel Based Apple Computers.
1 - Shutdown computer and disconnect all external drive Except the newly created Bootable Installer.
2- Restart and immediately hold the OPTION key until the Startup Manager appears and choose the USB Drive.
3 - It will present options >> Disk Utilities >> View >> View ALL attached Drives.
4 - Choose the Upper Most Drive ( not the volumes indented and list below ).
5 - The drive normally is called Apple Media or Apple SSD - that is the drive to Erase and format as HFS Journaled with the GUID Partition Map.
6 - Once that is done >> backup out of Disk Utilities and choose install macOS.
7 - Follow the prompts and it may automatically reboot several time.
8 - Upon a final reboot - Setup Assist will present with the newer version of macOS.
