I don't know if I am missing some concept, but I do have "time aware" data set up in my ArcGIS Online space, I have set the "Time" configuration to use a time range that contains "start" and "end" dates, My data "Date types" are set as "Date"
When I configure the Time widget in Web App Builder, it looks like the time span is only looking at the "Start" times of my data and not the "end" times of my data.
Let me know if I need to configure my data differently or if this is a possible bug.
Beta site: ArcGIS Web Application
and see the attached word document for some screen shots of the configuration.