With the help of Brain, I've defined the self as something positive : What has the qualia of it's own nature and existence - It's not my fault if qualia do not exist XD! However, without qualia, by definition, sensible experience does not "feel like" anything. Without qualia, we are just moving lumps of flesh with no feelings.
I know qualia exist because they are the fundamental aspect of my experience. The stimuli that
cause those qualia, in the purest philosophical sense, don't necessarily exist in the form that I conceive them*, but the qualia are the most direct aspects of my existence. I can't say the same for anyone else, because I have no direct experience of their perceptions, but I at least know that my consciousness and experiences exist, in some form, because I have them. File that one under "I think, therefore I am". The fact that consciousness and experience exist in some form is a tenable starting assumption for any argument, then, because I have a certainty of it's existence which is actually greater than the certainty of anything that my senses tell me exists.
Moving on to your definition of the self:
"What has the qualia of it's own nature and existence". So here you are talking about something that is capable of processing data (existence) into some other form (qualia). In a sense, what you are talking about is something that is intelligent (only in the sense that a calculator is also intelligent). The thing is, that kind of rudimentary intelligence is
only possible as a social process between multiple entities. In computers, it's a social process between transistors, and in our brains, it's between neurons. The things that must happen in order for information to be processed require multiple entities to communicate with each other, plain and simple. Therefore, any definition of the self that involves intelligence requires some kind of similar setup. In fact, it is very difficult to come up with a function for the "self" that doesn't require some interactions between distinct entities in order to work.
This pretty much means that you need to retract your reasoning for why the brain can't be the self, since your entire argument for why the brain can't be the "self" is based on properties which I have shown that your "self" must also have. Consider the following:
1. Consciousness exists (we at least know our own consciousness exists). This means it is possible for
some thing to have the property of consciousness.
2. There is no way, even theoretically, to prove the existence of a consciousness other than the one we experience.
3. The brain is an intelligent object that is heavily associated with conscious experience.
4. Your definition of self involves an intelligent object that is heavily associated with conscious experience.
The main objection with regarding the brain as the seat of consciousness is usually that we can't prove that it has that property. However, nobody even knows theoretically
how to prove that something has the property of consciousness. Why then, must we invent an object to be the seat of consciousness? What is the property of said object that would allow us to prove it is conscious? If it is possible for something to be conscious
and that we could never prove it, then inventing an extra object to house the consciousness is absolutely arbitrary. We are better off looking at what evidence links any
known object to consciousness. All of said evidence points to the brain. While this doesn't conclusively prove the brain as the seat of consciousness, it makes it far and away the best candidate, as we have actual evidence to support a connection, rather than conjecture. Because your definition of "self" is so similar to the general definition of "consciousness", I think you really ought to consider that the brain might be the "essesnce of the self" after all.
*I have pointed out that the idea that our senses are totally deceiving us is plausible and logically coherent. This doesn't that mean there is any evidence that makes this the preferred hypothesis. Occam's razor generally leads me, and most others, to believe that it is far more likely that our senses are approximating the real world (with imperfect accuracy, of course), rather than inventing something that has no empirical analogue outside of our conscious experience. The point of mentioning that there is no
true certainty outside of our conscious experience was a rebuttal to Jack Jack's saying "hey I'm not saying it actually exists", not an attempt to reduce this argument to "how do you know anything is real?".