I have experienced problems using sparsebundles on APFS volumes, whereby writing ~50gb lead to a backup that had incorrect checksums (determined by running tmutil verifychecksums and seeing a failure).
I then switch to using sparsebundles on HFS+ volumes, and no more problems.
Worth mentioning that you are probably using sparsebundles that are themselves ...
To be able to install/upgrade a new OS without completely erasing the drive (do you have backups?) you will need to either know the primary administrator account password for the machine, or be able to reset them.
You are going to need physical installer media to be able to reset the passwords, and restoring an InstallESD.dmg file from the bowels of a macOS ...
You can just set a new admin password and then you're able to enter the password for cloning the ESD.dmg file (which you'll then perhaps not even need at all).
You can set a new password by doing the following:
1) Power on the Mac while holding down the Cmd key and the S key
2) When the Mac has booted into single-user mode, type the following commands:
Using Finder go to the folder where the icns file is located. Open this icns file using Preview. Once the file is open click on select all in the edit menu item. Once you've selected all, open a file in Pages and paste the image in the new or old file. This file now contains the image.