A great fall...

[obloquy]

In any case, I wonder if others have any ideas on this: Why Saruman failed, while
Gandalf succeeded in following the path of light?

You touched on it already. Olorin's success was a result of his character -- the character he had from the beginning. The Silmarillion says:

But of Olórin that tale does not speak; for though he loved the Elves, he walked among them unseen, or in form as one of them, and they did not know whence came the fair visions or the promptings of wisdom that he put into their hearts. In later days he was the friend of all the Children of Ilúvatar, and took pity on their sorrows; and those who listened to him awoke from despair and put away the imaginations of darkness.

Unfinished Tales provides this little bit of insight:

But Olórin declared that he was too weak for such a task, and that he feared Sauron. Then Manwë said that that was all the more reason why he should go...

Humility.

Olórin truly cared about the people he was sent to help. HoMe X - The Later Quenta Silmarillion Part 1 contains this interesting version of the above-quoted text:

In later days he dearly loved the Children of Eru, and took pity on their sorrows. Those who hearkened to him arose from despair; and in their hearts the desire to heal and to renew awoke, and thoughts of fair things that had not yet been but might yet be made for the enrichment of Arda. Nothing he made himself and nothing he possessed, but kindled the hearts of others, and in their delight he was glad.

Olórin was just awesome. He selflessly sought to enrich the lives of others.

You may not be familiar with these recent threads in which I examine certain aspects of Olórin's character: Powers of the Istari and Sauron the Eyeball.