My English is not very good, so I want to make sure that I understand you correctly.
As I understand, your definition of “self” is a “permanent essense” (similar to atman - fixed, eternal, innermost essential and absolute something). Since you now introduce a new “real self”, so please explain to me what is this “real self”? Is it the same or different with the previous self?
As you said “The problem is the sense of self, the experience of a self”, I understand this statement as the sense of a “permanent essense” or the sense of “a fixed, eternal, innermost essential and absolute something”. Now the questions are:
Have you ever think that you are a fixed, permanent essense? I never recall that I ever do so! So, am I free of delusion yet? Why do you think that you are a fixed, permanent essense while you can clearly see the changes in yourself and the impermanence of everything around you everyday?
Do you think that all of us believe that we are a fixed, permanent essense and we never die or change? Why do we have people who do not believe there is next life or rebirth?
Since the fixed, permanent essense is nowhere to be found, how can we experience that self? (As you said the problem is the experience of a self).
As you said “The purpose of the Buddhist path is not to eliminate a real self”, so am I that real self? What can that real self do? Is it permanent or impermanent? What is this real self for?
As you said “a selfless person” is not an arahant. You said the purpose of the Buddhist path is to correct a delusion of self (sense/experience of self). However, if a person is selfless how can he/she has delusion about self? How can he mistake something with something that do not exist in him?
Moreover, if I did not study Buddhism or religion, I do not even know what is atman, and never think about the so-called “permanent essence”. Do you?
Sorry for having a lot of silly questions. Thanks for your response and patient.