Do you actually intend to use other github features? (apart from git hosting itself) If not, then what exactly are you trying to achieve by putting your code there?

- as you said git is relatively easy to maintain locally (at least compared to the standard customer-facing stuff)

- there's no difference in reliability / backup security really imho - unless you trust your local hardware less than "some random host in the cloud"

- not sure what your code does / is, but if you don't publish it, you don't even have to think about information leak (apart from standard host security)

- does the hardware cost play that big role for you? you didn't mention any other gain

- tbh, I'd be more worried about someone discovering that you host your public services' code online and starting to look for security holes just for fun - normally you don't need to think about the security of some one-off utility that you commit, but when it's online, it can tell more about your internal arch. than you want to show

So if you have a dedicated host for git and want to get rid of it, sure - even the private hosting will be cheaper. But is it more important than the other issues.