My comments below are for Apple iOS mobile devices. The quick answer is no, you can't use the Fastmail app or website on small mobile devices (such as any size iPhone) to access that screen. But you can use an iPad.
There appear to be two different mobile device Fastmail displays. This is true for both the Fastmail iOS app and the website (viewed using the standard Safari built-in browser).
- Large mobile devices (such as the iPad 9.7 and larger models - I don't have a smaller iPad, but I assume all iPads fall into this category) --- These show a display similar to that shown with a regular PC browser which includes the following items on the Files screen:
My Files
Trash
-------
Files Usage
Sharing
Websites
Access logs
- Small mobile devices (such as iPhones, even the largest iPhone 7 plus) --- These show only these items (as you found) on the Files screen:
My Files
Trash
The four missing items all still use the old style pages. They are essentially inherited from the Classic interface. It's my understanding that Fastmail has waited to update those pages since these were considered to be less important for the average user and they were also considering possible improvements to the features on those pages. The old Classic style pages can't be easily adapted to the small mobile screens, so we must wait until they are updated to the modern style. I'm also frustrated by the delay, but since I have an iPad Pro 12.9 inch I can easily use the pages which are missing from the iPhone display.
Bill