I think they should, but also offer Dualshock 3 compatibility and require that all developers include Dualshock compatibility to each of their games if at all possible. I'd prefer to play using a classic style controller if given the option to. The reason I think they should is based off both the Wii U and rumors of the NextBox using a tablet style controller. By including a Vita in every console as a controller, more of the gimmicky third party games will have the probability to be multiconsole or ported over to the PS4.
Everyone already knows the Vita desperately needs sales, and by doing this, Sony would kill two birds with one stone. Not only would each PS4 sold give them a Vita sale, but it would also allow them to compete with Nintendo by having the same gimmicky feature. So gamers could play PS4 with their friends at home, and then whenever going somewhere, take their Vita with them and play their Vita games on the go. Since Sony is including most first party games with the cross buy feature this would push the value of cross buy even further because everyone would already have both systems to take advantage of it. Also, if they were to push harder to get third party developers to include cross buy in their games by offering a bigger percentage to sales, this could persuade more people to go towards the PS4 version when deciding what system to purchase their multiconsole titles for.
Also, by integrating Remote Play between the PS4/Vita from the start, Sony could have more games compatible with it and even include it as a feature on the games product listings of which games work with remote play,for those people who are specifically looking for those that work with it.
When it comes to functionality, the Vita works fine, and the back touch panel is an alternative for L2/R2 by some chance one of those rare games runs out of buttons and requires everything, including the touch screen.
Here are some games that already have those same gimmicky features that we see on the Wii U. Nothing that specifically interests me, but just an idea on what PS4/Vita compatibility could do.
LBP2/Vita PS3/Vita connectivity. http://www.youtube.com/watch?v=x5SuLzHRo0Q
Sly Cooper Thieves in Time connectivity. http://www.youtube.com/watch?v=wTAX23oCPxE