Upgrade then restore, or the other way around? I'm in a situation where I simply cannot upgrade my Mac – the installer says that my disk cannot be used as a startup disk. This probably has to do with me installing dual-boot partitions, but I'm not sure.

I've decided to just wipe the disk and restore from backup. In the process, I'd like to upgrade to the most recent Mac OS version as well.

My question is simply: what is the most reasonable approach – upgrade the OS using a USB stick and then read back from the backup disk, or the other way around: restore from the disk, then run the upgrade?

Thanks.
Apple's Migration Assistant does a great job at importing data from previous installations or backups. Therefore, I suggest you save yourself some time and do a clean install of the version of macOS you want to finally use and then use Migration Assistant to recover your Apps, profiles, etc.

As suggested by @SteveChambers:


back up
create a bootable USB installer or use internet recovery
reformat the drive
install macOS, 
restore from backup using Migration Assistant