I'm not sure I fully understand. So you are saying that if you publish a workbook to Tableau Online, referencing a published Data Source (TDE) that is on a refresh schedule, that the workbook will not show the updated data after the refresh of the data source, until you manually refresh the page on the workbook?
if this is the correct understanding, think of it this way... The visual construct of a Tableau view or dashboard is the result of a query of the data source. What is returned from the data source gets cached and the images rendered. Periodically the data source has to be re-queried to ensure what's being shown visually is "current" with the data source. This can even happen with a live data source.
Many customers who deploy dashboards in a "shop floor" or "control room" environment (Big monitor visible by people in an area) will utilize a broswer plug in that can be set to refresh the browser view automatically, in order to overcome what I understand to be your concern.
Not making a recommendation, but one example of such a browser plug in for Chrome is "Easy Auto Refresh". Easy Auto Refresh
Hope this helps,