This is pure speculation.
What if “Red-nosing” is how Santa’s workers feel whenever an elf or a reindeer starts sucking up to Santa.
Rudolph is the brand new reindeer on the sleigh team and he spends most of his time telling Santa how great he looks, how Mrs. Clause’s cookies are the best, and how the North Pole may be freezing but, “Wow, boss, you sure make it ten degrees warmer when you step into the room.”
So everyone else says,
“Gee, we’ve seen some red nosers around here, but if you were to catch a look at Rudolph’s red nose, you would even say it glows.”
But it works. Because Rudolph gets to lead the sleigh team that night. And all of the other reindeer promise that Rudolph will go down in history is the one with the reddest nose of all.