There is the canonical QA How can I download an older version of OS X/macOS?. In addition, I have included two methods which are suited to your situation.
Method 1
Download an available version from 12.6.1 to 13.2 to the host Mac. Install the chosen version in a virtual machine guest on the host. Oracle and VMware offer free software. Boot from the virtual machine guest and use the command below to list available downloads.
softwareupdate --list-full-installers
Note: Guests can list more versions of macOS installers than the host can. Hosts will only shown macOS versions compatible with the host. Guests do not have this restriction.
From the guest, use the softwareupdate command to download an available installer application which is compatible with your MacbookAir6,2. Transfer the application to the host Mac. Follow the instructions given at Apple's website Create a bootable installer for macOS to create the bootable USB flash drive installer.
Method 2
Use of of the answer from the question How do I create El Capitan installer on a Catalina (or post-El Capitan) installed Mac for use on USB boot installer? to create a Sierra, El Capitan or Yosemite USB installer.
By using this method to get back to Big Sur, you would have to update your Mac. Recently, I have noticed that the Big Sur downloads from the App Store do not necessarily include all the software needed to install Big Sur. These types of downloads can not be used to create a USB flash drive Big Sur installer. So while this method may get you back to Big Sur, you may not be able to create a USB Big Sur installer by using this method until your MacbookAir6,2 is actually booted to Big Sur.