Why is it profitable to stop the cycle of rebirth if it is not "me" who is reborn? Or maybe to ask differently - to whom it is profitable?
The Four Noble Truths talk about suffering. That there is suffering, the cause of suffering, the cessation of suffering and the way to end suffering permanently.
It's not about the cycle of rebirth.
Please read this answer to understand what suffering is in Buddhism. It's about discontent and unsatisfactoriness.
Also, please read this answer and this answer to understand what rebirth is in Buddhism. It's the rebirth of individuality and not really the rebirth of the individual, since there is no soul or permanent consciousness in Buddhism.
Excellent logical question however the question ultimately has no relevance because it merely represents the false ideas that have arisen in Buddhism for the last 2000 years or so. In summary:
- The original scriptures refer to two types of teachings: (i) moral/mundane/kamma teachings for worldly people; and (ii) transcendent/supramundane teachings for enlightened people, namely:
And what is right view? Right view, I tell you, is of two sorts: There is right view with effluents, siding with merit, resulting in acquisitions [of becoming]; there is right view that is noble, without effluents, transcendent, a factor of the path. MN 117
- The moral kamma teachings that refer to various states of being that follow from certain types of kamma (actions) are never impersonal. Such teachings refer to the same "being" the reaps what it sows, such as:
I saw — by means of the divine eye, purified & surpassing the human — beings passing away & re-appearing, and I discerned how they are inferior & superior, beautiful & ugly, fortunate & unfortunate in accordance with their kamma: 'These beings — who were endowed with bad conduct of body, speech & mind, who reviled noble ones, held wrong views and undertook actions under the influence of wrong views — with the break-up of the body, after death, have re-appeared in the plane of deprivation, the bad destination, the lower realms, in hell. But these beings — who were endowed with good conduct of body, speech, & mind, who did not revile noble ones, who held right views and undertook actions under the influence of right views — with the break-up of the body, after death, have re-appeared in the good destinations, in the heavenly world.' MN 4
The transcendent/supramundane teaching that all things are not-self (anatta) are never found together with the worldly mundane teachings about "kamma".
Its like the teachings found in kindergarten compared to the teachings found at university. You don't find teachings about algebra, calculus and rocket science in kindergarten for the small children.
To conclude and answer your question, when you act in an ordinary worldly way, there is a sense of "self" behind that action. For example, if there was no sense of self (for example, no "I want to have sex"); there would be no performance of ordinary action. Since there is a sense of self behind the action; there is also a related sense of self behind the results of the action (for example, "I hate my girl who left me for another guy & I am now feeling hurt").
Therefore, it is profitable to stop the cycle of "rebirth" ("cycle of getting burned by self-centred actions") because it is profitable to be without the pain & suffering of doing misinformed careless self-absorbed things.
To the extent that you are reborn at all, to that same extent you are reborn from every moment-to-moment. With this in mind, ask yourself: "Why is it profitable to stop the cycle of rebirth from moment-to-moment if it is not 'me' who is reborn?"
Let's break this down into some piece-wise questions:
- Do you accept the premise that you are reborn from moment-to-moment?
- If yes, then would it be profitable to stop it? If so, why/how?
- If no, then what is reborn from moment-to-moment? Anything? Nothing? Is it "me?"
Maybe think of the above and if you come up with anything let us know! :)