The benefits seem unclear to me, especially how this solves the shared-state issue. Is it just that each "widget" calls the same API "manager" and propagates the changes to every widget in the event of a change?

Isn't this exactly what Redux already do if you use actions to fetch data and put it into the store, with the difference that each "widget" manage filters and state based on the fetched API data? Seems like it could create a lot more code with less re-usage, since every widget has to manage this themselves.