A toddler sees no benefit in sharing a toy whatsoever, because he wants to have the toy and by sharing, he doesn’t have the toy.
An older child begins to see that sharing is valued by his parent. He’ll share his toy because he’ll get praised or rewarded by his parent. This is doing something for the sake of a reward. Some people’s religiosity is in this mode.
The older child may also begin to see the benefit in sharing because it opens the possibilities of trade: if I share my toy that he wants, he’ll share his toy that I want. This is also doing something for the sake of reward, and is the underpinnings of secular “morality”.
An even older child begins to develop empathy: I will share my toy because it will make him feel good (bein adam l’chaveiro), or it will bring closeness with my parent who values sharing (bein adam l’Makom), and that makes me feel good too. I think this begins to transcend doing something for the sake of reward, but I can understand your argument that it does not. Fine.
A wise child realizes that he wasn’t put on Earth to have toys. He was put on Earth to fulfill a purpose. He learns about his purpose, and sees that sharing is part of this purpose. He shares to fulfill his purpose. I think this is a pure form of not doing for the sake of a reward at all, because even if the child ultimately feels good because of acting in this way, it is only because he aligned his feelings with his purpose.