Single player games can keep elemental affinities because they don't have to have balanced classes; it's not the end of the world if your fire mage starts fighting some fire resistant monsters because they either have other party members still controlled by the same player to pick up the slack or they can have a harder time but still be acceptable because they're not competing with anyone else for that specific time frame. If a fire mage starts fighting monsters weak to fire, because they're not competing with other players, there's no fundamental balance issue.
For MMOs, it's a completely different ballgame. Elemental casters are either specialists or generalists. If they're specialists, they're end up either overpowered (and therefore desired more than any other DPS class) or underpowered (and therefore anathema to anyone looking to form an effective group), neither of which is a good thing. If they're generalists, they're once again underpowered (if the advantages are baked in and they end up fighting an enemy without any weaknesses) or overpowered (if the advantages aren't baked in and they end up fighting an enemy with weaknesses), and they further penalize any player that doesn't have an encyclopaedic knowledge of enemy weaknesses and strengths.
Another problem is that, in the case of generalists, you either end up with an inordinately large number of attacks or a boring rotation: you either end up with a 6 3 button rotations that each focus entirely on a single element or 6 6-8 button rotations that make you end up with 3-4 times as many abilities as we currently have.
As I said earlier on in the thread, you only get 2 of 3 between balance, elemental weaknesses, and manageability and, honestly, elemental weaknesses add the least to an MMO.
Also, if you look at MMOs instead of RPGs, you'll note that elemental weaknesses are a rapidly vanishing mechanic and, afaik, only exist in games that don't give a shit about balance, are out of date, and/or simplistic. WoW has pretty much done away with them completely, and TOR has split up its "elements" into "damage that cares about armor and damage that doesn't" which NPCs treat in pretty much the exact same way (the "penalty" for using the superior damage type is the same as the advantage that damage type has against NPCs).