How do I revive Macbook air stuck in Big Sur Install?

Try restarting in Safe Mode - Intel computer Restart and immediately hold the Shift key until the Apple Logo appear the release. For Apple Silicon M1 - shutdown, then hold the Power Button for 10 seconds and follw the prompts to start in Safe Mode

Safe mode will do a Repair Disk, clear cache files and only load Apple Software - so loads slowly - Normal. This may or may not allow the upgrade process to complete.

Trust, the computer had the 35.6 GB of Empty Space Required for the upGrade ?

The Other category is a potpourri of files which can include:

• System temporary files

• macOS system folders

• Archives and disk images (.zip, .iso, etc. - often found in the Downloads folder)

• Personal user data

• Files from the user’s library (Application Support, iCloud files, screensavers, etc.)

• Cache files: browser, Mail

• Mail messages & attachments

• Fonts, plugins, extensions

• Safari reading list

• iTunes backups

• Crud resulting from jailbreaking your iDevice

• Game data

• Saved data files

• Call history

• Notes

• Media

• Voice memos

• Other files that are not recognized by a Spotlight search

• Media files that cannot be classified by Spotlight as a media file because they are located inside of a package

• Files created and modified by other user accounts on your Mac.

They can be located anywhere on your hard drive.

The files that you have control over are located in the Documents, Downloads, Pictures, Music and Movies folders.  You can use either of these two free apps, GrandPerspective  or OmniDiscSweeper, to find the largest files on your drive so you can determine if they can be deleted or moved to an external HD for storage.  

Note: you can empty the Downloads folder after the apps and/or updates that were downloaded have been installed or applied.  Many users have found a couple of Gigabytes of files in their Downloads folder which are no longer needed. 

To start with - suggest taking the very good advise from @Old Toad. This my allow the creation of additional required space to allow the upGrade to Big Sur.

That said - IMHO - Apple should not have being selling the 128GB SSD configured this way. That is way too small to do anything reasonably.

Even if you can get Big Sur installed - the 128 GB is going to be a real struggle to maintain the good practice of always having 15% the Total Drive Capacity as Free Space. Example 128 GB Capacity x 15% = 19 GB Empty Space at all times.

Do not suggest this except if not other options.

Create a USB Installer of Big Sur and use that to boot into and Wipe the Entire Drive including the Recovery Volume. That would remove everything and no do overs - period. Then, Install Big Sur and very very selective Applications.

Understood and Yes - advise and method supplied by @ Old Toad will definitely help to get some space back.