I'm not religious either, but a better analogy for a believer would be telling someone to pray or sacrifice a goat to stave off scurvy.
If you really believed some nonsense, sure you'd want to not keep it to yourself, and to tell others how to live. But that wouldn't actually be a good thing, despite the believer believing it was.
You twice used examples where a "believer" has (at least observational) evidence to support that belief.
Some of organized religion has roots in that (some of the Jewish dietary laws have practical basis), but much of the prescriptions of organized religion is preference-based rather than evidence-based.
If you had a cure would you just keep it to yourself so you don't "tell others how to live"?