Urm, I would state categorically that this is the most fun you can have with the same console on a shared game experience
The story catches you up if you have not played Diablo I or Diablo II and no it is nothing like watching a movie halfway through.
My wife and I have played It extensively and while she did play Diablo II she is more concerned over looting and killing things than what the actual story entails
The console version is also specifically designed for pickup and play functionality and allows 4 players on the same console.
On this topic, if you can get your hands on Rayman Origins and Rayman Legends, which also both allow 4 player shared screen gameplay you should have some child friendly gaming action that your whole family can enjoy.
On the platformer space, Ratchet and Clank All For One is awesome and the same thing goes, up to four players, one copy of the game, one console
You'd have to check availability of these titles on the console / consoles you have though as I think Ratchet and Clank may be a PlayStation Exclusive
EDIT:
See what you are potentially looking at is to maximize screen usage while cutting down on the amount of consoles needed, therefore what you are looking for is shared screen co op, which does not have split screen duplication of player viewpoints.
This allows for co op gaming even on smaller Televisions.