Is it possible to have the story map advance/cycle through the bullets or tabs automatically?
It is not possible in the standard application but you could do that with custom development.
We have no plan so far to implement that in the application.
If that's an option for you, there is everything you need on GitHub Esri/map-series-storytelling-template-js · GitHub to know how to advance to the next bullet or tab. The user interface would be on you.
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