"You don’t disappear in the sense that you cease to be as an individual mind, you are always yourself - that’s a tautology. What you lose is the sense of some identity that’s separate from what you ARE in this very moment."
This assumes that there's any concept of "you" that exists independent of your thoughts whatsoever. I think you're right that some Buddhist thinkers believe in this kind of essential awareness underlying conscious thought that you're describing, but others would say there is literally nothing underneath. "The self is an illusion", "all is emptiness", etc. If you believe in those ideas, then you have no awareness independent of conscious thought because there is no you independent of conscious thought.
The self as an illusion though refers to the concept that we as we exist now is ever changing and can’t be kept as some identity as that identity is necessarily not who you actually are. Our memories are not real - just shadows of an experience, and our aspirations are fiction. That does not however mean you don’t exist. You do exist, and as you move beyond the basics of mindfulness and releasing of attachments our role in a broader human experience becomes crucial - loving kindness (metta), and other aspects, are what are suppose to fill that void left by detaching from self and what isn’t. You move from being obsessed with self to becoming a force of goodness in the world by virtue of your existence. This couldn’t possibly be the case if the goal were nothingness and that’s it. This is where karma and other concepts come into play.
But to my understanding the idea of nothingness being some objective in Buddhism isn’t the case - but it’s often described as such because that state of pure awareness without encumbering thought and attachment in many ways to an unpracticed person feels like nothingness. After all, the awareness is silent, even if it is where all thought and feeling spring from.
Finally, awareness isn’t that moment you snap back to thought. You’re always aware. We just tend to be primarily aware of our thoughts and emotions. We walk around in a haze of the past and future and fiction as the world ticks by around us, and we tend to live in what isn’t rather than what is. You don’t disappear in the sense that you cease to be as an individual mind, you are always yourself - that’s a tautology. What you lose is the sense of some identity that’s separate from what you ARE in this very moment. You aren’t a programmer, you aren’t a Democrat, you aren’t a XYZ. You are what you are, and what that is changes constantly, so can’t be singularly defined or held onto as some consistent thing over time with labels and structure. You just simply are.