This thread is such cringe. Mostly because "self awareness" already has a definition and it's nothing at all to do with your incorrect definition in the OP.
As for actual self awareness (and not your redefined variety I'm not even going to touch because it's just flat-out wrong), it does require a certain bare minimum intelligence to be self-aware, so it is a measure (although a small one) of intelligence. People who are retarded enough (we're talking very extreme) lack self awareness, although most people have self awareness to an extent. You are aware you exist. That is literally what self awareness means. Awareness of self. Don't go redefining terms.
Also what you are terming "common sense" may or may not be sense at all. It's hard to say without knowing exactly what it is you consider common sense. The problem is nearly everyone considers themselves to have common sense but others to not, and it boils down to if people think the way they do or not. Well, some of those people are correct and others are incorrect, but they all think they're correct. Someone's "common sense" might tell them you walk around barefooted because there's no sense in wearing shoes because they never wear them, and that person might think all shoes-wearing people lack common sense. Most other people would think the no-shoes person lacked common sense. See what I mean?
As for claims of "it's only book smarts". Nah. No such thing. Either someone understands a concept or they do not. If they understand it it doesn't really matter how they came about that understanding, whether through reading books or through personal experience. If a "book smarts" person lacks understanding it's probably because they haven't read enough books on the subject. And, no, skimming doesn't count as reading. Also if you don't comprehend what you're reading, start over, try again, doesn't count until you do understand. If it's too hard, start out with more elementary books on a topic and then try again. Etc...
