Do you really want the strategic implications of your proposal?
You would not only strongly discourage players from attacking,
you even would strongly discourage any player from being the first to checkmate anyone, because according to your suggested rules, this would automatically trigger the remaining two players to be forced to team up against him?
Are you sure you are understanding the strategic implications of the current rule incentive system, and are you sure you are aware of the strategic implications of what you have suggested?
Is your aim to make this a game where nobody dares to start attacking, and nobody dares to be the first to checkmate anyone? Is your impression really that the majority of players will want to play slow and passive games rather than interesting active attacking games?
Very simple:
What do you think?
If you insist on the score system, the just consider the fourth 'suggestion'. This could be useful in 2v2 too.