I think this article suffers from a poor definition of "self". I believe the "self" is an illusion created by mind. It has no substance, and no existence when the brain is in "selfing" or "minding" as you call it. This lack of substance means there is no "there" there.
The reason I posit the definition of self is poor or incomplete is because all the arguments you make are arguments I would embrace as evidence that the self does not exist independent of the brain and body that produces the mind which generates the illusion of self.
Your title says "The self is not a fiction". Okay, what is it then? It's not anything physical. I think we agree on that. It's not created by God, nor is it a "soul" that implies a continuation or existence independent of the mind that generates it. This doesn't leave many alternatives other than to conclude the self is a fiction, a fiction created by mind, and this self ends when life ends.
What are the characteristics of the "real" self, which must exist if the self is not a fiction?
I don't contend that the illusion of self is not "real". It's as "real" as any other idea or concept in my mind. It exists because my functioning brain creates mind (I can't solve the hard problem of consciousness, so don't ask me how), but the "self" that my mind creates to explain its function doesn't exist outside of my mind.
As a scientist, you appreciate the need for the testability of your suppositions. How exactly would you prove the "self" exists? What test could you devise that could be independently verified by others? The "self" is rooted in subjective experience, and such, it's beyond science's reach. We can see the results of its activity, but as Thomas Nagel pointed out, we can never know what it's like to be a bat. We can learn everything knowable about a bat's behavior, but we will never know a bat's subjective experience of reality.
Indian meditators came to believe the self was an illusion 2,500 years ago. The Buddha expanded this idea and made "no self", anatta central to his teachings. The illusion of self is the primordial error, the deepest root of our ignorance. From this illusion, all other illusions emanate. The central teaching of Buddhism is to meditate on the unreality of the self and experience the "emptiness" of all phenomena. Hume came to basically the same conclusion 2,000 years later.
