The git integration is internal to Sublime; it doesn’t rely on anything external to work. I would think that since you’re seeing the indicator in the status bar, the git support is working for you.
If you open the Sublime console with View > Show console, you can enter sublime.arch() into the input widget and the response will be the architecture. For example, after doing so the console should say:
>>> sublime.arch()
'x64'
Note that icons only appear next to files that have a git status of some sort (added, modified, conflicted, etc). Does the repository that you’re testing with contain any changes? If the repository is clean, the sidebar won’t display any indicators at all.
Is the repository that you’re testing with your home directory? (not in your home directory, but literally your home directory).