Run only users are only available when you are starting the Flow from a Flow mobile button or from a selected item in a SharePoint list. Other than that you'll need to export the Flow and have the other users import it and create their own connections to have it run in their context.
But there is another likely explanation for why its not working. When a Flow is started from a button in PowerApps it automatically runs in the context of the user running the PowerApps. The first time they run it they should be prompted to authorize all the connections, including the ones used by the attached Flow. If the PowerApp and Flow connections where added to or changed durring development after the two were wired together this can lead to a disconnect and the wrong authorizations are created. If that is the case then you need to remove the Flow from the PowerApp, and then re-add it and wire up the button again. That usually fixes the issue.