Sounds like you will need to clean install everything via Internet Recovery Mode or build a USB flash drive installer using another Mac to cleanly re-install Big Sur.
Hold Shift-Option-Command-R while powering on until you see a spinning globe on a black background. Connect to WiFi and it will download the Internet Recovery into RAM and then you can completely erase the internal disk using Disk Utility. It should install the macOS version that originally shipped with this MacBook Air or the newest available version after that. The first user account you create during the startup wizard will be your admin account.
Then you can upgrade the OS to Big Sur using this articles instructions. Special links to the App Store to obtain hidden older macOS releases such as Big Sur because only Monterey will be offered by default in the App Store.
How to get old versions of macOS
https://support.apple.com/en-us/HT211683
Alternative is the download Big Sur using the link above to another Mac. Then create a bootable USB flash drive of 16GB+ by following this article: https://support.apple.com/en-us/HT201372
Hold Option key when booting with the USB Flash drive on the MacBook Air. Then erase the internal disk completely and install Big Sur.