Yes, poor old Rudolph was hated and kept from the reindeer games. Just like the fat kid always gets picked last for the playground team. And the teen with pimples never gets the girl.
Part of the message is that life is not fair. And you seem to have taken away that -- and only that -- message.
But how about the rest? That even the pimple-faced fat kid, who nobody wants to play with, can have a marketable skill and "shine" in America?
The message is pettiness is OK. Even Rudolf’s father is ashamed of him. Part of the problem is that the other reindeer are basically a nameless rabble. So it doesn’t portray them learning they were wrong, they just accept him once Santa (who also didn’t like him until that nose seemed useful) does. That really is the ONLY message: be liked and get liked.
There is no “rest”. And frankly your attempt to find one goes back to the basic problem of the story. Who cares about marketable skills? He’s looking for FRIENDS, he’s just a kid looking to participate with other children, he’s not looking for a job. The story winds up justifying the bad behavior, in the end they were right to hate him. It’s just plain not a good story, it sends the exact opposite message it purports.