I could have sworn they did this exact thing for one of the solo adventures and I wondered back then why we didn't get access to the hero in constructed play.
Yeah, it was Rastakan's Rumble solo adventure. It was a nice feature, that kind of resembles the process of picking a class for a new character in wow. However, it only featured a troll hero. What the op suggests - creating a seperate portrait for each race and class, looks very cool on paper, but falls short when you realize how difficult it would be.
First of all, most races have access to a limited number of classes. For example most of the alliance races don't have access to the Shaman class (except draenei, dwarves and some of the achievement races). Same goes for Paladins among the Horde's races - only Blood elves and Taurens have access to it. I know it might not seem odd for many hearthstone players, but seeing a worgen shaman or forsaken paladin would be ... at least very weird.
Then, there's the issue of Demon Hunter, which hearthstone already messed up lore-wise on many levels. In WoW (and generally in the canon lore) technically a member of any race can become a Demon Hunter, but in reality, almost only Night elves and a minority of Blood elves actually become Demon Hunters. It's not because of some form of "discrimination", but rather a natural result of the fact that Illidan is (or was depending on what you think) a night elf and the whole idea of "vengeance through sacrifice" is only understandable, if you are an elf yourself. In WoW (i belive) we never meet a DH of a different race. Sure, there are members of other races among the Illidari (like Lady Vashj, Akama etc.), but they're not Demon Hunters.
I didn't intend to write such a long comment on this topic, but as a big WoW lore guy I am just pretty sure OP's idea, as fun as it sounds, would make no sense and only make other lore fans as myself mad. (For real, that murloc demon hunter card almost made me stop playing hs xD)