copy the entire current blockchain to an external HD or SSD (whichever I have available), but first I delete the outdated backup
This is how you lose your data! Imagine this: you delete your backup, and right after that your main drive fails. Data gone.
So there's nothing I can do to optimize how backups are handled, right? I've already considered that possibility.
If you want to update for instance an offline version of Bitcoin Core, rsync would work. The first time, copy everything, and after that, only update the data.
Play around with
rsync --link-dest= a bit. Something like:
mkdir /path/backup/todaysdate
rsync -avh --link-dest=/path/backup/yourlastbackupdate /home/user/.bitcoin /path/backup/currentdate
This makes your backup grow maybe 6-8 GB per week, plus 12 GB for
chainstate. The latter will still consume a lot of disk space if you do frequent backups.