'Stripping away' is just division in another form. You're dividing experience into that which your stripping away and that which you're keeping. You keep doing this division until you get to the qualia. I think you can keep going with this. Say you look at a red dot at noon for ten seconds. There's no individual 'redness of red', there's the red you saw for the first second, the red you saw for the second second, and so on. This just leads to your next point though...
> Yet, we intuitively feel that there is something above and beyond those aspects of the experience.
Right, intuitively, we feel there is something more to the process of experience. I get that. Yet many things we conceptualize don't work in an intuitive fashion. Why do we require experience to?
I certainly think that there's more to experience and consciousness than meets the eye, I just wonder if examination of the process and trying to individuate experience as qualia is actually helping towards that goal.
I think the reason rejection of qualia is the mainstream point of view right now is because most everyone believes consciousness works like a computer. Sure, there's this world of hardware underlying the operating system, in the same way that there's a biological brain underlying consciousness, there's still this entire world we can reason about and play in and theorize about even despite this. There's no reason to go "Oh, we're just robots" because much of our behavior can be deduced from our neurology. Dennett says this, but it's clear he doesn't really feel like a robot and is being a little tongue in cheek.