That's not neccessary true. Take for example, in the case where the proof shows that resulting ciphertext would be indistinguishable from a random function, so

That's not neccessary true. Take for example, in the case where the proof shows that resulting ciphertext would be indistinguishable from a random function, so the adversary would have 1/2^n (with n = key length, 128/192/256 etc, ) chance to decipher the text by guessing the key uniformly random. And he also has another 1/2^n chance to break the system, by us picking the null key (randomly). That would result in the chance of 1/2^(n-1) On the other hand, the proof shows that leaving the null key out of sample space means the adversary would have 1/2^48 (arbitrary number) to distinguish the ciphertext from a random function, which means that it's "insecure" in the context of the proof. It seems to me that this notion of secure would still be stronger than our common sense idea of "secure".