Kushina Uzumaki’s dying wish in actions of Obito Uchiha, Kushina and Minato had little time with their son. To save the lives of not just Naruto but the entire Leaf Village, the Fourth Hokage and the former Jinchuriki had to sacrifice their lives while also sealing the Nine-tails spirit in their son.
Already anticipating all the struggles an orphaned Jinchuriki would have in life, Kushina spent her final moments in the world asking the Third Hokage to take care of Naruto, to which Sarutobi agreed and assured her that he would take "full responsibility". However, Naruto still grew up living alone, had no idea who his parents were, and was never informed as to why people feared him despite most of the village’s adults being aware that he was the “fox kid”. In other words, the life Kushiha thought Sarutobi would give Naruto and the life the future Hokage had couldn’t have been more different.
Although the Third Hokage himself never discussed the reason why he did not keep his promise to Kushina, one can deduce his actions were deliberate. Despite his involvement in the Hyuuga clan fiasco and the Uchiha clan massacre, the Third Hokage was far from being an evil person or an overly pragmatic leader like Danzo was. That's why the only reasonable explanation for Hiruzen not to have raised Naruto or given him special treatment within the village is that many enemies would be looking either for the Fourth Hokage's son or for one of the tailed-beasts Jinchurikis, meaning no one could have any idea of how important Naruto really was. That’s why Naruto was given the Uzumaki surname instead of the Namizake surname, and why no one who knew about Naruto being the fox’s host was allowed to discuss it.
Protecting Naruto from threats from outside the village made sense, especially considering the Akatsuki was already coming to be and this helps to understand some of why the Third Hokage did what he did. That said, whether Sarutobi's decision was worth it is debatable. Naruto had to grow up with the pain of not knowing anything about his parents, the loneliness of being an orphan with no other relatives in the world, and the hatred and prejudice of the entire village. Had it not been for Naruto's unshakable spirit and his dream of becoming Hokage, the character could have walked a very different path and Naruto could have become like Gaara: a violent, resentful Jinchuriki who hated everyone around him. Considering that by the time Naruto was 13 the Akatsuki was already after him, Sarutobi's plan might have been redundant after all.
Hiruzen Sarutobi failed to give Naruto the life Kushiha had asked for, but the Third Hokage at least ensured the boy's safety until he could become a Genin. Sarutobi also placed Naruto under Kakashi's care, granting the Fourth Hokage's son arguably the best Jounin a team in the Naruto universe. Like all of the Third Hokage's other decisions, keeping Naruto in the dark about his background was a controversial call, one that will forever be discussed.