Is it possible to keep the full iOS twitter integration (tweet from anywhere in iOS, get notified on lockscreen/notification center etc.) and uninstall the official Twitter client and use TweetBot or Twitteriffic instead?
I found out recently in iOS 6 you don't need the official Twitter client installed nor the Facebook client. I use Tweetbot for Twitter, and I am still able to use the built in Twitter features for iOS without having reinstalled the official Twitter app.
Notifications are generally handled by the individual applications through the notification centre and therefore you don't need the twitter client installed, since your preferred client will handle these.
It seems that the link for this in the settings page is only to make you aware that you need another client to read actual Twitter/Facebook messages and to get all functionality.
iOS provides native authentication and tweeting functionality. The reason for the recommendation to install the twitter app is due to the fact that iOS does not support reading tweets or notifications natively.
I found this out by accident setting up my iPhone 5. I have since removed the official app from my iPad HD and everything still works.