Save files from corrupted SSD

I think you'll need to rethink how to recover the files here. Ubuntu LiveUSB by default won't access the Mac file system so it won't show you the files you are trying to gain access to. Yes, you can make it work but it's a bit late here for that. Here's how one person did it: Ubuntu installation on USB stick with pure EFI boot (Mac compatible) I my self have never done it so I'm not sure even this works to the point of being useful here. It also won't gain access to FileVault2 encrypted drives or folders.

OK what to do??

You'll need a second Mac system to either connect to your MacBook Air via Target Disk Mode or remove the SSD from your MacBook Air and place it into an external case and connect it to your good Mac via USB.

• Use target disk mode to share files between two Mac computers
• OWC Envoy Bus-Powered USB 3.0 case for Apple MBA 2012 SSD

The other possibility here is to create a bootable Thunderbolt or USB drive and then use it to boot your system so you can salvage the files and as well see if you can save the SSD (reformat it & install a fresh copy of OS).

• How to create a bootable macOS Sierra installer drive
• How to set up and use an external Mac startup disk

Sadly, you'll still need access to a second Mac to prep up either the Installer drive or the external bootable drive. Frankly, you'll likely need both!