OK, so I posed the question, and I found the answer. (Of course I had to find it myself, I just don't have people who regularly read my posts and do research for me!)
The situation is this: I use cPanel on my web hosting server. I use the cPanel backup tool to regularly backup my home directory (includes my web files, mail, etc), and my MySQL databases. I love the fact that you can use it to backup to a remote FTP server, and I do that on a fairly regular basis.
The major drawback is that you have to remember/schedule to perform this backup manually. This becomes especially difficult if you have multiple cPanel accounts. It would be amazing if you could just schedule the cPanel backup to run at regular intervals, perhaps with cron. Well, although there's no option for that in cPanel, the script below will allow you to do exactly that!
This backup script includes SSL support. This isn't necessary if you run the script on the server for which you're generating the backup; but the SSL support is very important if you're running the script somewhere else to connect to your cPanel hosting account.