No, you have never tried to examine what the supposed problem is. You have supposed that there is a problem and then jumped to trying to explain th alleged problem.
what way is the consciousness "unified"? I dont think it is.
Do you mean that the "first person perspective" seems to be just one?
To know what it means to be unified, think about what is not unified in your mind. You know that there are memories that are currently not being accessed and experienced. You know there are other minds experiencing colors that you aren't ecperiencing. Or if you are in a deep thought, you may not be conscious of the blue sky in your sight (they call this selective awareness which is strange and fascinating if you haven't looked into it).
Anyways, clearly there is something that is unified about your consciousness and something that is not/discontinuous, namely the unconsciousness and everything else.
Of course they correlate. Being processed, refined etc does not remove the fact tgat they are correlated.
The activity correlates just fine, but only the activity, and only certain activity at that. But it doesn't correlate structurally. For me or for you, there are only certain parts of your own activity that correlate to consciousness, but none of the inactivity that occupies so much of the brain correlates. That structure just doesn't fully correlate to your "space" of conscious existence. In this respect, there is a sort of dual existence that filters out any inactivity that isn't known to correlate.
(Read to the end before picking apart pieces to argue against)
The main problem is that you wouldn't be able to predict such a thing as a unified consciousness by knowing what we know about all of the parts. Every other known physical phenomena can pretty much be predicted using the standard model and the laws of physics (reducibility).
The other part of the binding problem is the stuff inside this "unified stream of consciousness" can be compared. For example, if you see a truck, you know it's not a car. You know at least 2 things about 1 thing. They call this the "aboutness". And you know that you know 2 things, not just 1 thing and then 1 thing. The concept of the aboutness requires a unity of understanding from multiple pieces of information.
To understand what I am trying to explain - while holding judgement untill the end of the post - think about the consciousness as the part of the cruise ship we only get experience as a customer and the brain as the whole ship. For the vacationers there is a sort of perfect world, but in the background there are people on and off the ship that contribute as well as the machinery behind the scenes; all this is never seen.
But unlike the consciousness, the boat has a spatial dimension that we can follow from the customer space to the spaces that have less appealing aspects of the boat. There are no nonexistant places because we have a 3d space to navigate through. But when I am visually emerged infront of a white wall, where's the unconscious stuff working behind the scenes? Why can't I get to the "workers scrubbing the cooking grills"? And why can't the people looking in go from the "machinery" to the ship's presentation for the customers? Obviously I can't go where there is no brain activity; it just doesn't exist for my conscious space. What does this say about our supposed 4d space and time?
And you can't solve the problem by just saying that the consciousness is the activity because that is the same problem just restated. Why does something else exist just for certain processes (does it have a function/purpose), what is this conscious stuff (the ontological question) and how does matter without conscious properties form a new property that we call the consciousness (question of origin/reducibility)?
The why, what and how questions I ask are rhetorical because there are no obvious or agreed upon answers to them.