Trouble reinstalling OS on 2015 MacBook Air

The problem, as I see it, was the 2015 MBA drive was erased without having a Bootable Installer of macOS 12 Monterey in hand.

A MBA without any Operating System will possible revert to trying to install the Original Version of OSX / macOS it Originally came installed with

That would be and sadly OS X El Capitan was revealed on June 8, 2015

You could try Internet Recovery Mode

Option-Command-R: Start up from macOS Recovery over the internet. Use this key combination to reinstall macOS and upgrade to the latest version of macOS that's compatible with your Mac

It will present options >> Disk Utilities >> View >> View ALL attached Drives. 

Choose the Upper Most Drive ( not the volumes indented and list below ).

The drive normally is called Apple Media or Apple SSD - that is the drive to Erase and format as APFS with the GUID Partition Map.

Once that is done >> backup out of Disk Utilities and choose install macOS. 

Hopefully, it will offer macOS 12 Monterey 🤞

Follow the prompts and it may automatically reboot several time. 

Upon a final reboot - Setup Assist will present with the newer version of macOS.

You can use Setup Asset and point it to the Time Machine Backup Drive. This is a One Time offer.

Thereafter you will need to use Migration Assist 

You are very welcome

We are pleased to help and even more pleased it all worked out

Happy computing