Things I wish still existed:
Play by Committee games.
I originally found this in Space Empire IV. Basically, all (human) players are part of the same empire. They are system governors, generals, mercenaries, trading companies, etc. And they all kinda role play within this system (There are other AI controlled empires that the players can interact with).
Barring something that intricate, it would be nice to play Civilization or something as a council? Or delegate (human) players to different sectors or fields of the state.
I guess the board game equivalent of these would be the MEGA GAMES ( http://www.megagame-makers.org.uk/ or http://megagamesociety.com/ ).
But I kinda don't wanna set aside an entire day for something that can be super (socially) exhausting so I am not going to a megagame anytime soon...