The God Object needs to be referenced at least transitively by the wrappers you hand out, and its API surface needs to grow constantly to encompass any possibly security-sensitive operation.
Dependency injection doesn't help here much, at least not with today's languages and injectors. The injector doesn't have any opinion on whether a piece of code should be given something or not, it just hands out whatever a dependency requests. And the injector often doesn't have enough information to precisely resolve what a piece of code needs or resolve it at the right time, so you need workarounds like injecting factories. It could be worth experimenting with a security-aware dependency injector, but if you gave it opinions about security it'd probably end up looking a lot like the SecurityManager did (some sort of config language with a billion fine grained permissions).