With a mouse, you use your entire arm and wrist and all fingers to control a device that has an arbitrary amount and speed of movement. You can set the sensitivity to anything you want. You can move the mouse as fast or far as you want. You can, if you are a ridiculous heathen, also accelerate the mouse response non-linearly based on movement speed.
With a flick of the wrist, you can express either very subtle movements or very large ones very quickly. You can move your aim over a tiny pixel or do a 360 degree spin in a tenth of a sceond.
With a gamepad, instead of using your entire arm/wrist/fingers, you just use your stubby little thumbs. Instead of having a square foot or more in which to resolve your intended movement, you have about a 1.5 inch circle.
The idea that there’s even a debate here seems silly to me. The mouse is so obviously far and away a more precise way to control that it’s silly to argue otherwise. The only things that sticks are better at are games that require a constant-rate input - like flight sims or racing games where you might want to maintain “turn left 50%” for several seconds.
And even so, this is a false dilemma. People paint this as “pc is kb+mouse, console is gamepad, both have stuff they’re better at” but that premise is faulty. PC is whatever the hell you want to use. You can play a game with a mouse, gamepad, driving wheel, joystick - ffs, I bet I can beat console gamers using gamepads in FPS games using only the throttle on my X-52 pro. I use my xbox 360 gamepad more on my PC than on my xbox. If you want to use a gamepad for FPS games, go ahead and set your PC up that way. We won’t laugh at you too much.
Anyway, I would probably pay triple for any game that let me play against xbox livers and it would be worth every penny. It would be such a comically delicious way to take out my rage on them while they helplessly tried to flick their thumbsticks in my direction. DELICIOUS XBOXERS.