‘Debate’ does seem to have strong connotations of a rhetorical fight. ‘Discourse’ has been the preferred word among philosphers and other circles that discuss this sort of thing.
"Discourse" is a much broader term and does not imply formal structure, opposition, merit evaluation, or that advocacy of particular ideas is associated with particular speakers.
IMO "Debate" is absolutely the right term for what the advice suggests, which is to reframe a conflict situation into a situation where there are norms of fair argument and expectations of equal access to present opposing views, with the hope of evaluating the merits of said views.
The problem with debate as it exists today in the english sphere is that it refers to a sport where teams try to argue for views they don't even hold regardless of what they themselves even think about it. Due to the idea that regardless of the actual merits of an idea debated, either team needs to have an equal chance to "win", the whole "evaluating merits" part has in more recent times been deemphasized.
If I'm understanding you correctly, it sounds to me like you think the current use of the word in the USA is restricted to one narrow activity of debating societies (formal tournaments where neither side is, a priori, invested in a particular idea and in fact doesn't even know which side of the debate they'll have to represent). This is just one of many things debating societies do, and yes it does have a sport element as you and others suggest.
But, based on my experience (and I have quite a lot with debating societies in both the USA and the UK) I don't think this one activity has come to dominate what "debate" means in the USA. I would be surprised if the average citizen thought in this way.
So I wonder what makes you think this activity has come to be the meaning of debate in the USA, as opposed to just one thing debating societies do?
P.S. purely as an aside I think there may be misunderstanding about the value of people advocating "views they may not even hold". This is only characteristic of tournaments, and is a feature not a bug. Since you don't know until quite late which side you will represent, it forces you to be prepared to do an equally good job representing either side. This is, in my opinion, an excellent mental exercise. At the moment the debate happens, you are arguing something one-sided you may not believe — but in the preparation for the debate you are carefully evaluating the best arguments for both sides.
You're certainly right about the long tradition behind 'debate', and yet, it does still carry the connotations I mentioned earlier, to the extent that one would likely have to say "let's a have a fair and honest debate about X" in order to temper some of the unproductive tendencies that tend to arise in a debate.
Anyway, with (Plato's) Socrates, dialectic doesn't seem to have been two people candidly searching for the truth either - usually it was Socrates expertly pretending to do that, while actually showing some expert just how much he didn't know about his chosen subject.
For me, dialectic and derived words mean Hegelian dialectic. That's 99.99% where I've heard the words used.