A way to put it:
The cycle of Rebirths refers to the cycle of the rebirth of the 'ego'. That is what is said to reincarnate as a self-perpetuating illusion into cycle after cycle of suffering due to the ego being, essentially, interconnected karmic chains of desire. It is desire that is illusion as Truth clearly reveals that there is nothing that is 'real' enough to desire.
But Energy goes on! Energy is not a substance but an action. It is Karma. Just as objects with mass carry momentum, desire also carries momentum. It is simply the progression of states from imbalance to balanced. It is the law of Entropy at work. So even when The Arahant sees the truth and can understand all things clearly, He reincarnates. But the ego does not. The Arahant is born in his next life as someone who naturally has little or no ego and effortlessly reaches enlightenment again in their youth, and carry on in cultivation of his karmic bodies until either reaching Union and/or Self-Annihilation.
But it cannot be said that the Arahant has not escaped the cycle of rebirth. What is reincarntated? It is not the ego. It is not anything that can be called a 'self'. The Arahant reincarnates not as a deluded ego, but simply as the strings of karma that continue after extinguishing the Great Confusion. He reincarnates as a light in the world of darkness. He guides others, even before re-enlightenment, because without the ego, our focus of our karmic momentums naturally turn toward the Divine Desire of the Reconciliation of All Things.
After enough lifetimes, the focus on the karmic momentums has caused such purity of awareness that we must call that bundle of energy something other than Arahant. The string of energies has united so well with the Karma of the World while at the same time in complete Union with the Divine Desire, that it can teach the WAY in any set of terms, ideas, or religion that is useful to the moment. We call this a Buddha.
Don't worship the Buddha. There's nobody there