Sometimes our beliefs depend on a few pieces of easily identified evidence, but often they depend on the balance of a large quantity of (potentially forgotten) evidence colored by our experience and aggregated using intuition.

Wholly agree. Further, even recognized good counter-arguments may not sway immediately if the idea being replaced has been integrated into a larger networked structure of (possibly accurate) beliefs. It takes time to tangle out a new network that is free of the false idea. I find that it usually takes an attack from several different directions to let go of a wrong idea that is well-integrated.

In fact in ordinary conversation no one even tries to use Aumann-style negotiation, and from my perspective this seems completely rational.

I've noticed this too. People simply state lots of beliefs and people are just expected to nod if they agree or disagree. If they're not sure whether they agree or disagree, then they can ask questions along the lines of, 'why do you think that?'. So that the Aumann process only proceeds in marginal cases, when someone is likely to be swayed. This seems efficient.