Nothing really stops a bard or minstrel from being a profession. And the only line of separation between classes and professions are gameplay. Otherwise, Blizzard literally doesn't differentiate the way you imply. By your logic, 'Alchemist' shouldn't be a profession either because its a Hero class in WC3.
There's no hard line in the sand defining Enchanters must be a profession while Evokers must be a Class. The Mage class wouldn't be any different if it were called an Enchanter or Evoker instead. It'd still be a spellcaster all the same.
I mean Blizzard literally changes the definitions of existing class archetypes all the time. In WC2, a Death Knight was a spellcaster unit equal to a Mage. In WC3 and WoW, it is a heavy armored melee combatant. In WC3 Alchemist was a melee combatant who drinks potions, heals with chemicals and throws acid bombs. In WoW it's a profession.