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?
You need more storage and it's more labor to setup, both of which could be impractical depending on the systems you manage, but the central service would have no access to anything confidential. Also, integration between the central backup and the variety of local hosts would be greatly simplified, with no file locks, database backups, consistency issues, backup window coordination, etc.
EDIT: Also, the granularity of available recovery points might be limited, at least from the central service. Block-level backups (and the associate recovery points), for example, could be implemented locally but might be hard to implement on the central service.