Okay I had the same basic experience
I was running MacOS Mojave and it was running well no complaints but I don't like to get to far behind so I ran a time machine backup first unencrypted.
Then ran the update to MacOS Big Sur
I got the same error "An error occurred preparing the software update"
Tried recovery mode but that did not work as it would not unlock the drive with the correct password (this is troubling and makes me question file vault ) I also tried using my apple ID to unlock the drive but this did not work as it could not unlock the drive.
Tried Safemode but that would give me the dreaded Circle with a line in it
I figured out it was most likely due to lack of storage space. Then found this post.
This is what I did.
I used a MacOS Mojave usbkey installer (I made a while back)
I install MacOS Mojave to an external drive (Nvme SSD usb c drive)
after successfully booting to the external SSD drive
I then was able to unlock my internal drive (the password worked) and I remove 50 GB of data
Then shutdown disconnected the external SSD
Then rebooted to the internal drive and the install of Big Sur Finished
I know I should have looked at the requirements prior to upgrading.
I would hope in the future the Installer for all MacOS Updates checks to make sure the system meets the needed space requirements.
Hope that helps.
Brian