They don't go for anyone vigorously, except possibly the human player(s).
Which is actually their biggest weakness in winning the game. They often attack the human player who will be able to organise and beat them (due to the faulty power graph making them think they can win... with billions of their rubbish ships vs fleets that they can't beat) when they have puny AI players neighbouring them and some easy minor race for the picking that they could snatch up to forge a huge empire and a real problem for the human players late game. They should definitely tear into each other more and the warlike AI, I guess, should be able to make early inroads into their lore enemies.
We could even encourage human players to play the same way and have to embrace the lore of the game with penalties. Eg as the Iconians you have a 50% happiness penalty if you don't recapture Iconia from the Yor by turn 100... It strikes me that recapturing their homeworld would be somewhat of an obsession for the population...
Obviously all races would need something in balance that would encourage the AI and humans to play in a way that is more aggressive and fits the story. Of course teaching the AI to not just capture some random world then wait for it to culture flip back would be a good start (saw the Snathi engaging in that fun game vs a very puny Torian empire)!