Is the save data for Angry Birds still there even if the app isn't?
I believe you backed up from Recovery, correct?  If that is the case, then the backup should contain everything that was on your phone before the backup -- apps, saved games, etc.  So restoring the backup should have restored Angry Birds and its save data, as long as that was present before the backup.

Titanium can back up the app, the app's data, or both.  It does require root.  Non-root users have to deal with Recovery backups, which are not as flexible since you can't restore only bits and pieces -- it's all or nothing.  Of course, the SD card is easy to back up (plug it into a computer) and some apps may allow you to export data.

You'd have to ask Rovio why they chose not to use a database, but I'd guess it's because the save data is so simple.  If you open the file it's almost human-readable text.  Even easier to back up than a database.