Since you’re interested in this kind of DIY, approach, I’d seriously consider thinking the whole process through and writing a simple script for this that runs from your desktop. That will make it trivial to do an automatic backup whenever you’re active on the network.
Instead of cron, look into systemd timers and you can fire off your script after, say, one minute of being on your desktop, using a monotonic timer like OnUnitActiveSec=60.
Thinking through the script in pseudo code, it could look something like:
This would pull the back from your server to your desktop and, if the backup failed, use a service such as ntfy.sh to notify you of the problem.
I think that would pretty much take care of all of your requirements and if you ever decided to switch systems (like using zfs send/recv instead of rsync), it would be a matter of just altering that one script.