If the DM eliminates Paladins because he doesn't like them, it may seem arbitrary. But making him run a campaign in which you've "forced" him into letting him run a Paladin is ultimately futile. Its probably going to suck, and he's probably going to stop running the campaign.
Do you feel that a player who is "forced" to play in a campaign where there are Paladins, even though he hates Paladins, is also going to "suck" at his contributions to the game, and probably drop out?
When I was a kid, my younger brother used to insist that if my mother made him eat foods he didn't like, he would throw up because they were so gross. Now, I'm willing to agree that its objectively mean to make a kid eat food that he hates so much that it actually makes him throw up. Its a good reason not to eat something! Except that it wasn't true. He threw up because he'd intentionally dry heave over and over and over until he got a little stomach acid, then proclaim that he'd proven his point. No one was fooled, and we ignored his antics until he quit doing it and just ate what my mom cooked.
That's the impression I'm getting from this style of argument. If it were genuinely true that the presence of dragonborn or whatever in a campaign were so offensive that they genuinely ruined the ability of someone in the game to even enjoy running the game, then I suppose they should be removed from that campaign, even sans any other reason. The player who wanted to play a dragonborn would probably be ok with choosing something else if choosing a dragonborn genuinely wrecked someone else's enjoyment of the game. He'd way his desire to play a dragonborn, and compare it to his friend's distaste for them, and say, "Woah! He totally hates dragonborn! His misery will far outstrip my enjoyment of my own character, so I will choose something else!"
Except that this whole scenario is crazy. Why is it exclusively DMs to have this problem? Shouldn't players logically have it too, at the same rate? And of course we'd see even more of it, since there are more players, right? But we don't. We just see DMs proclaiming that this or that would wreck their ability to enjoy the game so much that they wouldn't even be able to be interested in running the game well.
I'd posit that this is because DMs have the power to ban things, and players don't. So players roll their eyes and get over their dislike of dragonborn when their fellow player wants to play one, and just go enjoy the game. DMs don't have to do that, so they don't.
Its like a really big strong guy claiming that someone else's insult made him so mad that he couldn't help but punch them in the face. He might even genuinely feel that way, but if he were a smaller, weaker guy, he'd have learned to control his anger by now.
A DM might genuinely feel that a particular player choice is a campaign wrecker, but if he wasn't accustomed to having his whims become law, he wouldn't feel so bothered by his whims not being followed...