I don't think that's it - I can absolutely think of things that mix drama and comedy, including fairly juvenile comedy, and still manage to work. The Ace Attorney games, for instance. Or heck, The Order of the Stick. But I can't really call anything in Hearthstone drama - there's not nearly enough story being told here for that. One villain monologueing to to you about how doomed you are while you play a card game does not a dramatic moment make.
Whether Hearthstone could do drama or not, I'd say they've long since passed the point where they've set the game's tone solidly to "lighthearted comedy," so trying to do a villain who's serious in any way which isn't itself funny seems like a poor idea to me.
Haven't seen Yu-gi-oh Abridged, but I didn't really have that kind of reaction to the Lich King myself. I think you'd need to take his attitude up a few notches for me to feel like that's what was going on - have him get really petulant about you beating a boss whose deck he thought was great, perhaps, and rant about how you just got insanely lucky.