Though you already seem to have found your choice, I might point you to Ghost Commander. Similar in size (at least concerning the
.apk size), it is not only "gratis", but fully open source. Offering a dual-panel view (at least on tablets) and being able to access network sources (FTP and SFTP out-of-the-box, Samba/Windows Shares via addon), and on the device itself also can access all locations (not restricted to the SDCard), inclusive root-mode if wanted. You can have bookmarks for your most used folders, it has ZIP support, a built-in editor and more.
Ghost Commander Screenshots, presenting its capabilities (click images for larger variants)
**How does Ghost Commander meet your requirements?
- Runs smoothly on Android 2.2: Minimum requirement is Android 1.6
- Not occupying tons of space (few MBs only):
.apk size is almost exactly the same as your mentioned choice (775 kB)
- Provides a file browser, and options to open files from there: Sure it does. Even lets you edit text files, look into ZIP archives, and more.
- Less is more, as basic as it can get without losing functionality: Uh, that's rather opinion-based of what belongs to "basic"...
- Is gratis: Free and Open Source
For more alternatives, be welcome to browse my listing of file explorers, which has – where available – links to reviews and more.