Larian actually did something similar to the Star Wars Old Republic mechanic in the original D:OS with a "rock paper scissors" mechanic.
This mechanic was dropped in D:OS2. The trouble seems to be that they wanted you to have "freedom" especially in co-op with 4 players. They seem to have imagined ideal co-op as the 4 characters each abandoning each other to do their own thing. One consequence of this is that players have a lot of "freedom" to not participate at all
in important conversations your co-op partner is involved in, and you often miss out on key dialogue.
This is part of the reason why "fighting for control of the dialogue" isn't quite as easy to implement as it is in Star Wars: The Old Republic (where the whole party would be forced to participate).
I actually think the co-op was better in the first D:OS largely for this reason. It felt a bit more like you were actually co-operating (and arguing) a bit more (and therefore interacting with your co-op partner) instead of just having one person take the lead while everyone else watches.
That said, one thing that would also help is if you could listen to conversations without having to actually choose to listen to them. In D:OS2, your co-op partner could be talking to an NPC right next to you and you won't hear anything unless you click to listen in. This also meant you almost always missed out on hearing the first line of dialogue.
This all detracts from the single-player as well. In Bioware games, your companions would e quite talkative, but that's been stripped out a bit because of Larian's focus on co-op freedom.
Larian certainly needs to improve their co-op dialogue design somewhat.