You can try to examine the Big Sur install log located on the "Update" volume on the internal drive for clues to the failure to complete the install. You will either need to boot from an external macOS drive, or access the log from Recovery Mode using the Terminal command line, or you can put this Mac into Target Disk Mode to attach it to another Mac so you can access the installer log. Maybe you can figure out what stopped the installer and fix the issue. If you can access the Big Sur installer log, then post the last 50 lines here using the "Additional Text" icon which looks like a piece of paper.
You can also restore macOS from a backup made before you attempted to upgrade macOS.
There is a chance you can revert to the "local" APFS snapshot that the Big Sur installer as it began the upgrade process. To do this you need to boot into Recovery Mode and select the "Restore from Time Machine Backup" even if you don't use Time Machine. Select the "local" option and select a snapshot dated from just before you ran the Big Sur installer. After the system reboots you should have the system back to how it was before the attempted upgrade.
Always have frequent & regular backups. You should always make sure to have a good backup before doing any major system changes such as upgrading macOS.
