Where'd My iTunes Today Widget Go?
I'm not sure why, but your file icons, and other various things shown in the screenshots are not what I see on my system. I run a 10,1 model so perhaps its just my system being old and boring looking. If not, I feel like some configurations you've changed might be the issue.

I honestly suggest reinstalling. You don't need to wipe any data, just re-download the installer for macOS 10.13 High Sierra A (Or the most recent downloader for Sierra, you don't specify which OS you have) from the Mac App Store and start the installer. Installations and updates will re-write all System information when installed which should restore your notification center files completely.

I understand this is a cliche and over-suggested method of fixing macOS issues, but no one is going to be able to figure out exactly what went wrong without detailing everything.

If you don't want to do that, we can go through a whole series of commands to validate your system. Starting with codesign validation.

You can use the codesign command to check if system resources have been altered and are valid.

codesign -vvv --display /System/Library/CoreServices/NotificationCenter.app

Copy and paste into terminal for various Apple (or non-apple) applications, frameworks, etc, to check code signatures and for valid or broken resources.