If I understand correctly, the problem is when a participant in spoken in-person conversation is working from an assumption they think to be obviously true to everyone, and other participants don’t think so.
For written conversation, hypertext can help; the “obvious” info can be behind a link that people don’t need to follow if they agree it’s obvious. For spoken in-person conversation, I don’t see how there could be any solution beyond what you describe in the last three paragraphs. But it sounds like you intuitively think some further solution might be out there. Can you identify what feeds this intuition?

The linked video is now private. Perhaps quote them inline?