Until self awareness happened, human beings did not exist - they were just apes who could walk on two legs. And until an "AI" has self awareness it is just a computer.
Defining something as intelligent when it is not self aware is just meaningless playing with words. A world beating chess computer is no more intelligent than my alarm clock. It's just a machine doing a job.
Funny, you should mention "meaningless playing with words". You redefine three important concepts: "intelligence", "existence", and "self-awareness" in ways that just aren't relevant.
"Existence" is the most broken concept. I'm currently sitting in a chair. By that observation, I can determine that the chair exists. Since it exists, then it must have in your sense enough "self-awareness", which puts a really, really low threshold on the amount of self awareness and intelligence that a thing needs in order to exist. None seems to be the absolute minimum threshold here.
Similarly, intelligence is not the capacity for self-awareness. Instead, it is:
the ability to learn or understand things or to deal with new or difficult situations
If we are to speak of the narrow category of the things having to do with learning or understanding aspects of oneself, then I suggest that wisdom is a closer label than intelligence.
And of course, this brings up the obvious point that if we have a machine that can fulfill the definition of intelligence far better than humans can, even if it has absolutely no self awareness (say due to deliberate programming of that blind spot for safety, efficiency, or ethical reasons), then of course, it is more intelligent than we are.