According to the last bullet on this page, external drives containing time machine backup should be bootable and appear as EFI Boot on the Startup Manager screen. My MBP late 2011 is running lion 10.7.5. I formatted a new external hard drive to HFS+ encrypted by erasing the drive using disk utility, and its map scheme became GUID automatically after the erasing operation. After time machine created a full backup on the drive, disk utility even shows it's bootable on its disk info page. However I still don't see an EFI Boot option on the startup manager screen nor on the startup disk preference screen. I'm not sure if it's because my MBP is too old or I missed some important step. Any advice would be appreciated.
By the way, backup content is fully accessible in macOS Recovery so the drive itself should be fine.
By the way, backup content is fully accessible in macOS Recovery so the drive itself should be fine.