Welcome to the Apple Support Communities and thanks for asking your question about getting your Game Center information to sync with games you've previously played. I love gaming on my iPhone and iPad and I don't like to lose save data either.
While it's true Game Center has changed from a stand alone app to a setting and sign in with the Settings app, the functionality hasn't changed.
You should be able to go to the App Store to redownload any previously purchased game app, and as long as you're also signed into Game Center using the same Apple ID used the last time you were playing the game, the information should sync automatically. If you run into issues, you may want to contact the developer as some games have additional sign in requirements.
Please let me know that this helped you out.