This is a great idea - unfortunately, some of the servers I need to back up are feature-poor shared hosting accounts where it would be difficult or impossible to run a script that serves this purpose locally.
Do yourself a favor and stop using shared hosting services. A few years ago, you could justify it because of the cost. VPSes are so cheap these days, though, that it's silly to hamstring yourself with shared hosting. Digital Ocean is a good place to start, and they'll even pre-install LAMP for you if you want. $5/mo for your own box.
I've been planning to get everything off the shared hosting accounts for awhile. Is Digital Ocean a better deal than, say, AWS for hosting dynamic websites?
