There's already an auradin in D4, once the tower is released I'm certain the meta will be similar to how D3 uses all classes for different roles. I don't think it will be 4 speedy sorcs clearing a pit 70 as the gauntlet was. Even if it was the case and your gauntlet runner was a sorc for fastest clears then to compete means using leaderboard meta (sorc). Then when you go back to questing or other in game content you're left without the cool cosmetic you just earned. A lot of players have multiple characters, they don't just play one specific class. Some classes are good for farming where others are slower to farm.
I think you have some high hopes for this game mode. I think it's going to be a massive waste of time. Already said that you're not getting loot out of it. You're JUST going to be able to get a cosmetic. If they want it to have any notoriety, it would be bound to the character and the season. Regardless of what "role" you're implying they serve.