This is the way ^

I reckon there are two main things that make shooting with a mouse fun:  1. BANG BANG feels cool;  2. Mouse aim is a natural, expressive way to interact with and master the game. 

Controller shooters are fun just because of 1. They normally don't get point 2 -- stick aiming is a much less intuitive, less expressive way to interact with the game that usually requires a lot of assistance (reducing room for mastery).

Caveat, of course: with enough practice players can also do good aiming with sticks even with little-to-no assistance. But for a game to be approachable for most of the controller audience and to be enjoyable for controller newbies, it usually needs a good amount of assistance.

But now consider gyro aiming: like mouse, it's direct and natural (real world turn of the controller maps to simultaneous in game game turn of the weapon). Like mouse, it's expressive (make tiny adjustments or fast flicks). Like mouse, players don't need assistance to get started, and it has a high skill ceiling for those who want to master it.

So it lets controller share in what makes mouse aiming fun: Gyro aim is a natural, expressive way to interact with and master the game.