Consensus agreement would probably not be the goal of such a project, and there's no reason to expect or desire that (aside from the old idea that public reason leads to truth in political questions which by now seems rather silly).
If we were devising a government (or any other system) it would be productive to have consensus on some issues (who is the enemy) and not so on others (quantity of net redistribution).
Just as there exists P space and NP space it ought to be possible to categorize questions for which there exist right and wrong answers (universal absolutes), and then a separate space for 'spectrum' questions whose answers are a matter of degree contingent on the feedback from the environment.
Confusing these two types of questions when the answers are unknown is easy and can make you very wrong. I'll give a practical example that may be biased. Personally I believe moderate climatic change to be in the 2nd category* while biodiversity in the 1st category because a temperature can be ratcheted up or down (even if it took centuries or was expensive) whereas once a species is extinguished it is gone forever while the side affects are permanent. They are both serious problems but getting a category 1 question wrong makes the error baked in for the future.
You'll remember numerous occasions when the government tried to defy the laws of biology or physics and this never ends well. Something I believe most politicians do not comprehend is that not every 'policy question' belongs to category 2. Something I believe most scientists do not comprehend is that not every political question belongs to category 1.
> aside from the old idea that public reason leads to truth in political questions which by now seems rather silly).
A public debate could be seen as a method to factor in information from the world and limit the framing of the question so something can objectively be done.
Not so much leading to truth as increasing the likelihood of survival for the participants.
* Yes I'm aware temp change can cause extinction events, this is a toy example to illustrate the point.
https://en.wikipedia.org/wiki/Dialectic
The problem with such a project (which I think could succeed without solving it) is that it is not merely the case that people disagree on the facts.
They also disagree on the frame of reference. That may not be solvable.