FPS obviously KB+M. If that's not possible, gyro.
Almost everything else that I play, controller. There's a few exceptions like some twin stick style games where the mouse still feels faster to aim. Like Neon Abyss, I wouldn't play that game with a controller, despite having platforming, because it's much easier to aim and shoot around with the mouse. And it's easier to move around and attack in games like Undermine as well.
I have this controller. This is the Mobapad HuBen 2. This controller has perfect tweakable super accurate capacitive sticks, and it doesn't fucking matter. Aiming with the sticks is horseshit. It will be if they are perfectly accurate and can be set up in 1000 different ways. They're just not good for aiming. I'll play casual third person shooters with it and I still won't find it as good. I'd rather the sticks have the other type of "capacitive" which is the one that the steam deck has (you touch the thumbstick and it activates gyro).