This seems like a privacy setting, or some sort of notification setting, but I can't seem to find any option to reenable this functionality.
Issue at hand:
If it helps, I was on the iOS 10.2.1 beta before downgrading back to 10.2. And if it is due to some form of corruption from iOS 10.2.1b1 how should I go about fixing the issue? Should I do a full restore?
In your screenshot of your current settings you are actually on the correct page, you just hadn't scrolled down far enough. If you go back to 'Settings > Notifications > Messages' and scroll down to below the "Messages Options" section the feature you want to enable to fix your problem is called "Show Previews". Hope this helps! And if it does not fix the issue you're having could you post a new screenshot of what it shows below the "Messages Options" cutoff point in your original 'Settings' screenshot?