Just like you've mentioned, you can try the fsck command in Single User Mode to check and repair your Mac's internal disk. Maybe you don't give any command when booting into Single User Mode so it quickly goes to the normal login screen and you have no opportunity to run fsck.
Now, do the following step-by-step to run fsck to repair your Macintosh HD drive:
Step 1: Start or restart your Mac, hold down the Command + S key combinations together to boot into Mac Single User Mode.
Step 2: Some white text will scroll by quickly. When it stops, you'll see a command prompt at the bottom of the screen that says root#.
Step 3: Type the following command and then press the Enter key:
This runs fsck with the -f flag, which forces it to check journaled file systems like HFS+, as well as the '-y' flag, which automatically says 'yes' to any prompts that fsck may encounter.
Then, your MacBook will run a diagnostic on your internal drive, and attempt to repair any damage it finds.
If 'FILE SYSTEM WAS MODIFIED' appears on the screen, it means one or more problems have been found and repaired. Since additional issues may exist after a repair is made, you should rerun fsck until you see the message: The volume (name of your volume )appears to be OK. This means your hard disk has no problems now.
Step 4: Simply type 'reboot' and then press Return to restart your Mac after finishing fsck repairing.
Try to update your macOS to see if the "Failed to prepare the software update. Please try again" problem still exists. If not, congratulations! you can update your Mac to the latest macOS version. If still, your Mac won't update to Ventura, Monterey, or Big Sur, try to erase your Mac and reinstall macOS as you've already done. But this time, after reinstalling macOS, don't restore your Mac using TimeMachine backups.