You’re fortunate to have a Time Machine backup. How did you erase the drive?
My suggestion is to reboot into Recovery (cmd-R), run Disk Utility and reformat the drive again. Just select the main drive (I suppose “Macintosh HD”) and click on Erase. You will lose the data partition and have a single partition left. Select APFS format.
If successful, quit Disk Utility and choose to reinstall macOS Catalina. If the macOS installation is good, choose your Time Machine drive for the migration source.
By the way, if you previously installed Big Sur, even if onto an external drive, Recovery will probably choose Big Sur to install because of your current firmware version.