The agruement about shamanism being reconned for the alliance in BC (and later Cata) fails when you remember that the Draenei learned shamanism from the Broken.
Essentially when the Broken lost the Light, there were many, like Farseer Nobundo, who turned to the elements for answers. This makes sense since the Draenei did live with the orcs for a long time and relatively in peace until Gul’dan corrupted them, so the Draenei did have knowledge of the elements. And these Broken shaman taught it to their unmutated cousins, and Velen welcomed all the shaman into his people.
The dwarves becoming shaman makes sense because the Wildhammer had shamans, and they taught it to the Bronzebeards. Not to mention the Dark Iron have shamans due to their long connection to Ragnaros and the fire element in general.
Kul Tirans have shamans because they revere the ocean, so they have contact with the water elementals.
In general, the races that don’t have shamans are usually those who just never went out to revere the elements (such as SW humans, all elves in general, and gnomes).
Regular timeline orcs that don’t have priests because they’re a shamanistic culture and don’t have the reverence to generate the Light. Mag’har have them because they were taught by the Draenei and the Shadowmoon clan before the Draenei of their timeline turned into crazed Light zealots.
Like I said before, the only classes that make most the sense to be for everyone are the physical-focused ones (warriors, hunters, and rogues) and death knights. The former is because you don’t need any innate ability or even a specific trainer for it (as one can be self-taught) and the latter because you just need to be raised up by a Lich King after you die.
The special magic ones are iffy because of cultural beliefs and such. A draenei wanting to learn warlock magic will not find anyone within his own people to teach him due to the stigma against it (In fact, they may even lock him up for that). This isn’t like “kid dresses Goth and listens to heavy death metal to spite his uptight religeous parents”
Some races may not even have the apitute to learn said magics. There’s no prominent tauren mages I know of, so its likely they have no ability to learn the arcane (The Feltotem gained their warlock abilities from the Legion directly).
And the agruement “D&D lets every race be any class” falls when you consider this: D&D gives players absolute freedom in their character backgrounds and that they are freelance adventurers teaming with others. But WOW doesn’t give you that full freedom. You can argue about Exile’s Reach, but generally your character is an adventurer that is also conscripted to serve in your faction’s military. Your background is generally based on your racial background: if you’re a Draenei, you’re from Velen’s ship; if you’re a Goblin, you’re a member of the Bilgewater Cartel; if you’re a (SW) human, you’re from Stormwind