How did Apple re-install everything in a clean Big Sur install, without access to backups?

Solution 1:

Though iCloud does store more than just Wi-Fi passwords by default (assuming you're using it), it isn't the primary culprit. Unlike some other operating systems, macOS doesn't wipe everything when installing (or reinstalling) the OS by default. While I don't have a definitive list of files it touches (and it likely varies by macOS version), you can typically assume that it will primarily touch system-related files.

What appears to have happened in your case is that your initial "failed" installation actually copied over many of your personal files successfully from the Time Machine backup; however, for one reason or another, something related to the OS installation failed and prevented the drive from being made into a bootable disk. When you re-installed macOS after the fact, those files remained in place, and macOS simply installed the OS alongside them.

Given the fact that the initial Time Machine migration failed, the exact state of what was restored isn't entirely clear. I'd recommend wiping the entire disk via Recovery Mode, reinstalling, and then using the setup utility to restore from Time Machine. Remember to format the SSD as APFS rather than HFS!