All the team projects have that as an implicit element, don't they? In order to arrive at a solution, the team has to talk, divide labor, etc.
Or one person takes charge and steamrollers everyone else, or one person ends up doing the work because the rest can't be bothered, etc. You're right there's often some implicit element of it, but how to do it well seems to be rarely taught. At least it wasn't made explicit at my schools, and I've heard little to indicate it's been made explicit at my sons schools either. It seems pretty arbitrary how well people pick up those skills.
What we understand as debate today, is exactly the school adversarial debates that you're referring to.
The "cooperative" debate is what we call a conversation.
Throwing the word constructive in front of debate, does not change the adversarial nature of it.
But it'd be good if people got better training in "cooperative" debates, where the purpose is to learn and listen. I can't recall many instances where that was encouraged in school, but there were a few. All of them were focused on a subject, though, rather than actually teaching the skill of listening and debating constructively.
There's a space for both - sometimes you need to be combative -, but people switch to combative mode way too readily.
To your point on the time formulating a response: Online I find a good measure of whether I'm "listening" or gearing up to being combative is whether I end up significantly re-writing my reply while writing it as I process the comment I'm replying to, or if have it "ready and loaded" to fire back at someone by the time I hit the reply button.