Titanium Backup - Restoring manually backed up apps (TB, Google Drive)
Titanium Backup would ask you to restore the .tibkp file if you "share" it to it directly. Easiest way to do so is using a file manager and "tap" the *.tibkp – as Titanium Backup is associated with that file extension, you should be offered to share it that way.

As you confirmed, Root Browser is a file manager capable of that. Most others should do as well, though some might be "too restricted" in their functionality as you found out, e.g. the built-in file manager.