It is pertinent to understand just why the Buddha rejected nihilistic philosophy because that reason also has bearing on why he chose not to answer the OP question about Self by simply clarifying “There is no ‘Being’ either before or after the dissolution of the 5 aggregates - only the illusion of a ‘Self’ brought about by craving and clinging to impermanent, conditioned natural processes.”
Its not difficult - we, the unenlightened have managed it after all! So why does the Buddha never come out anywhere in the Suttas and just say that?
Though this succinct statement is indeed factual and true, it is IMO not a useful framework to adopt if we seek the end of suffering. It leads towards Nihilism, which is an intellectual morass.
Consider the implications of this view of ‘no Self’. If there is no Self within my aggregates, is it therefore correct to assume that that also necessarily holds true for the (apparently) sentient beings I encounter? Are all ‘Sentient’ beings perhaps products of ‘my’ Mind, generated the same way that ‘I’ generate ‘my’ Self? What is the problem in depriving another being (who might also proclaim ‘no Self’) of life? Is lying/ stealing/ sexual misconduct OK for one who doesn’t grasp at ‘Self’?
IMO, when it comes to practical application - Negation of Self turns out on close examination to be truthful but not skillful, Affirmation of Self turns out to be skillful but not truthful. Indeed, all questions of Self invariably turn out to be traps leading away from the primary goal - the end of Suffering.
This could, IMO be yet another reason why the Buddha refused to answer such questions regarding Self and encouraged us to set them aside.
Indeed, the Master pointed out to Kaccana that the Right View of the noble disciple is that
he does not take a stand about ‘my self.’
So, we must not leave this discussion with the thought
‘I might not be, and it might not be mine. I will not be, and it will not be mine’?
as that is yet another form of sakayaditthi.
Instead, we must reframe the thought as
It (Form/Feeling/Perception/ Choices/ Conciousness) might not be, and it might not be mine. It will not be, and it will not be mine.’
in which case, our opening statement should be carefully rephrased as
The experience of suffering (re)arises and ceases because the impermanent, conditioned natural processes of the 5 aggregates are clung to consequent to ignorance and craving. With the complete ending of Ignorance comes the final end of craving and clinging - the aggregates are eventually scattered like chaff, never to rearise, thus bringing about the complete end of suffering. The framework to achieve this is the 8 fold path.
Come to think of it, isn’t that what the suttas keep harping on about? That the 5 aggregates are ‘not - Self’?
PS I apologize if this view is all much too subtle hair splitting - it could just be papanca for all I know.
ADDENDUM : I came across this Mahayana sutta which says something similar
Now, the tīrthikas—those who hold views based on objects and who engage in concepts and analysis—went among the followers of the Mahāyāna…"Is there truly a self in the body or not? It would be proper for you to dispel our doubts.”
The followers of the Mahāyāna replied, “Friends, it should not be said that there truly is or is not a self in the body, because to say in this case that there truly is or is not a self is mistaken speech.