Why would you do this while a game is playing mid game. All those things like checking trophies, sending and checking messages you do when you pause the game. They could easily do this without the screen, by tapping the PS button for a quick menu, if the PS4 OS is fast and allows multi tasking like Vita the. It should be easy without a screen.
This is true.
Although I think video chat or multi-way video chat with friends while playing would work better with a secondary screen.
I think if Sony wanted to find the business case for including a touchscreen on the controller they wouldn't have to look far.
1) Augmented Reality seems to be a big area of interest for Sony going forward. They've shown interesting technology demos using Vita, they have talked previously about a further future where lightweight AR glasses will factor in. A home system with a screen on a controller + camera on the back would give them a big stepping stone toward that ultimate goal. Coupled to the power of a home system, I think you would see amazing stuff from Sony wrt things 'breaking out of the TV' into your living room, as seen through your controller, that would gracefully evolve up to AR goggles in a few years time.
2) As I mentioned before, it would let them deliver Playstation content from every platform to their new system.
3) Cloud seems to be a big deal for Sony going forward. Depending on what else is in the controller it could potentially open the door to huge new business model opportunities coupled with that. People have talked about contracts subsidising an expensive box next-gen. If a customer has a good network connection and you don't need to give them that expensive box, you could go further still and offer them a controller with a pre-payment on a pay-as-you-go system. PS4 - starting at 'Free', no contracts.
As I said before though, it would also raise some other questions. But I think Sony could definitely see the case for doing this.