That Refresh button should work with Live Connections.
So if you make your connection Live that should work. ( but then it is not helpful to go with Live connection most of the time due to performance issues )
I do have one Live Oracle DB report which refreshes with Refresh button
--------->Connecting to Data Sources
First of all, some data sources will not allow extracts—they will only allow live connections. Obviously, if that's the case, you don't have to make a decision: use the live connection.
That easy choice aside, there's a long answer to this simple question. As you experiment with different approaches and learn more about the many variables that have an impact data performance, access, freshness, and the ins and outs of specific databases, you'll formulate an answer that works for your users in your organization.
However, we understand that you need to get your users connected to data today. So in this section we provide some guiding principles that you can use to make sound data access decisions as you roll out your shiny new Tableau Server.
Refresh button in the first screenshot is for refreshing the report/dashboard and not the extract data source. It will only come in handy when user level cache memory is legging behind the extract datasource (and off course for the live connection). For example, in your case if user opens a report before 7.00 AM, user is likely to see old data through the day due to cache memory (I am assuming your admin has selected "Refresh less option" on server settings). So in other words, 'Refresh' button on the report/dashboard will only help to get in sync with tableau data source.
To address your issue, only option I can think of is change the schedule to update your data source periodically....let's say every hour. Now, every time user opens the report and hit the 'Refresh' button they will get most unto date data. I hope this helps.