twitter could not authenticate you

Twitter for the Mac has been withdrawn and will no longer work.

It has been withdrawn from the App Store.

Sorry, but you'll have to use a different app or the website. Only the official app has been affected afaia.

Try setting up another admin user account to see if the same problem continues. Also try the Safe Mode.

Safe Mode - About El Capitan
also Sierra

Safe Mode Startup – El Capitan. also Sierra

If it works in the Safe Mode, try running this program when booted normally and then copy and paste the output in a reply. The program was created by Etresoft, a frequent contributor. Please use copy and paste as screen shots can be hard to read. Click “Share Report” button in the toolbar, select “Copy to Clipboard” and then paste into a reply. This will show what is running on your computer. No personal information is shown.

Etrecheck – System Information

Does this also mean that the notification center widget is useless, since it seems to use the authentication from the official app, which is no longer supported as you said?

Without a working associated app, the Notification Center will not work.