Way back during the school bussing brouhaha in Boston, there was one activity that seemed worthwhile - strong opponents and supporters of bussing were put in a room without the media and left to their own devices. As I recall, they came out of the room hours later - not with agreement, but with a better understanding of where the other side was coming from, their goals and dreams, their fears and concerns. In other words, they had a conversation. This is not something even remotely possible at a national level, it could not be on TV, it could not be lead by politicians.
If it happened today, the conservatives would come out alive and the liberals would be dead. And on that day, we'd have taken one small step towards restoring the Republic.