This interpretation is valid. But I recently learned to see it a different way.
If you'll humor me, please consider this. Since Santa knows if you've been "bad or good," he knows the other reindeer have been bullies to poor Rudolph. And, while a red glowing nose is cool, it's not a useful fog light. It's just not.
So Santa "uh oh!" had an emergency where, for the first time ever, the fog was going to be too thick all over the world to deliver presents?
Nope, he set up Rudolph in a position to "lead" his peers in a situation that maybe needed a little help but was not, in any way, a true, worldwide magic-assed Santa emergency. Santa knew how to guide his reindeer to accept each other. The story of Rudolph was not about Rudolph doing something to prove himself. It was about recognizing a Rudolph in need and helping him rise to the occasion to bring him closer to his peers in a way that could heal division.
Rudolph isn't about how to triumph as a Rudolph. It's about how to be a good Santa.
(Edit: For everyone who already thought this was obvious in the story, thanks for letting this Rudolph have his epiphany anyway.)