If we are talking about the franchise as a whole, then they already do that. Just not in the games. Those indeed feature a more blank slate that allows players to choose their dialogue options and mold their characters, but if you look at the manga and the anime, you get very distinct personalities for the main characters. A prime example is Yu from Persona 4. Most fans consider him the most unassuming of the protagonists, but in the anime, he’s a complete menace, going totally off the rails and hilariously over-the-top.
And that’s sort of the issue, in my opinion. An anime or manga follows a set route, but a game does not. If you inject too much personality into the blank slate of a game character, you risk losing that blank slate. Although I agree that more organic dialogue would be nice, not just with the social links but in general, however these games are already massive as they are, so I understand why they might choose to keep it on the simpler side.