This would imply that the confidence interval around the coefficient in a logistic regression is not a very useful concept, which I don't think is true.

That it is a little different. There you are estimating a continuous parameter (which happens to be interpretable as a probability) and it makes sense to have a probability distribution over that.

But if you are talking about whether a single discrete events will happen or not, a single number (the probability) already fully captures the uncertainty about it.

There is a lot of missing information in their probability of binary events presented. Presumably they polled N forecasters and are presenting an x/N prediction. The fact that each forecaster is estimating in a continuous space and then binarizing their result means that a lot of information has been lost.

To look at an extreme example… were all the “yes” votes 95%+ certain and the “no” votes just under the line 49%? Or was it more like a bunch of no votes at 49% and a bunch of yes votes at 51%?

Binarizing forecasts necessarily discards information. Aggregating a bunch of binary predictions into a percentage does no recapture said information, unfortunately.

It looks like they are simply providing the summary data to their multiple choice survey questions. Still, CI or SEM would not apply (there is no SEM for 80% of forecasters said 'yes'). These graphs are literally just telling you the percent of responders who picked a given answer to the corresponding question.

edit- the longer I browse their website for the exact methodology, the less impressed I am with this group. The "Introducing the Superforecasters" section is so cringe.

https://goodjudgment.com/about/