I recent played around with the SharePoint App on multiple mobile platforms: iOS, Windows Phone 10 (yes, I still have that...). I connected to our SharePoint Online environment (the App also supports to connect to SharePoint 2016 on premises). Although the (navigation in the) UI needs some time to get acquainted, the SharePoint App definite has potential to become a low-barrier anytime/anyplace entrance into SharePoint. Access through the hamburger-menu the content of the SharePoint site collection, pages, list items. And document libraries? Well, yes and no: trying to navigate and open any document library from the menu resulted in an error message about 'missing SharePoint license'; which I'm pretty sure that I possess... However, navigating via 'Site Contents' to the document libraries does work, presents the library view, and allows to open the administrated documents in/via the App.
From user experience perspective this is undesired and also inconsistent behavior. On the one hand, trying to open it from the end-user logical approach via the menu results in an error that you cannot relate; and on the other hand when you navigate via alternative path in the App the document library contents can be accessed.
The question is then foremost why it does not work via the hamburger menu? I derived the answer to this when I tested the SharePoint App on an iPad, as then in the screen I observed a message about trying to start OneDrive to operate the document library from within the App. And current we have not yet enabled OneDrive in our Office 365 tenant. So it not that I do not have a SharePoint (online) license in our tenant, but the OneDrive license.
Bottom-line: productive and user-friendly usage of the new SharePoint App, requires that OneDrive is enabled for you as Office 365 end-user.
