Now that communications and combat status updates are out of the way, what else does a group provide? Loot! Or, more generically, reward sharing.
Personally, one aspect of design I’m eager to change is level based progression, but that’s a separate issue. Reward sharing actually comes in two forms. The first I’m going to call inherent. These rewards are things like experience points or deed flags where simple membership in the group (and proximity to the event in most games) garners you a share. The main reason for this sort of structure is to prevent exclusion of “support classes” from rewards. If your group is fighting a group of monsters and you are the healer and during the entire kill of one of them you cast no spells, the group structure ensures you get a share. Obviously, more complicated “cast spell on person who fought” award trees could work most of the time, but I specified “cast no spells” for a reason. You are a vital part of the group, they need you, but it just so happens that for sixty seconds during one fight no one was hurt enough to require healing, so you didn’t. I suppose you could get even more complicated and add to the award tree anyone who cast a spell on someone who engages the monster within the last X minutes, but that could easily bog down the system with keeping track. A better solution is actually to remove rewards from the act of defeating a monster, at least for experience and move it to quests/tasks.  A number of games, most notably World of Warcraft, have already begun moving in this direction where grinding experience points fighting monsters is far less rewarding that fighting monsters that contribute to a quest that will yield a large chunk of experience as a reward.  Even though, group membership is still used to assign the quest flag (the kill of a rat for a “kill ten rats” quest).
At this point, we could start looking into different methods of awarding flags, such as the award being an area effect so that any player character within range gets the flag whether they contributed or not.  Each of them valid, and each can be done, but every method, even grouping, has exploitable elements, so the issue becomes which exploitability are you more comfortable with and to begin looking into ways to combat it -like logging out people who are AFK too long and trying to eliminate users who “macro”. Of course, the main reason some people don’t participate in combat is because combat design around things like the holy trinity (tank/healer/dps) encourage it, but that is a separate issue.
Its beginning to look like the current design of the reward structure, how players progress, and how combat functions in many MMOs (primarily the Diku style ones) are very dependent on the group structure and trying to remove that group element is going to require thinking the whole thing over from the ground up.