If everyone involved (players/organizers/audience) is having a good time, it doesn't matter. That being said, using a controller in a FPS lowers the max skill ceiling a great deal. You lose out on three main functions, one of which is partially solved.

First, accurate aim speed is slower with a controller. When people think about the limitations of a controller, this is usually the first thing people think of. This is why successful console FPS games have aim assist built into them. It's done so transparently that players don't even notice the aim assist. For example, in previous CoD games I've played (it's been a few years), when your reticle turns red and you ADS, it automatically snapped the aim to target even if kept moving. There's also things like bullet magnetism and the obvious target follow that pulls your crosshair as the target moves.

Second, fast changes in direction are essentially impossible with a thumbstick. Strafe dodging incoming fire is essentially impossible. First, you can't change direction fast enough, second the aim assist helps the person shooting at you to hit you no matter what you do.

Third, aiming and jump/crouch/reload at the same time is quite difficult. All those buttons are normally handled by your right thumb. This is the one that's partially solved, either by putting buttons on the back of the controller or by the claw hold.

These limitations of a controller force very specific map and game mechanic designs. You don't see complicated mechanics becoming successful on console games. For example, this

The video is in Tf2. The guy on the right who gets the kill is playing Pyro, the guy on the let is playing soldier.

Rocket Launchers (Soldier) can be used to rocket jump. You shoot at the ground, jump and crouch at the same time. Where the rocket lands, how your strafing and the timing of the jump crouch determines where you go flying in the air.

Pyro's have a flamethrower. The secondary fire on the flamethrower reflects rockets. You can aim where those rockets go.

The pyro aims the reflect a little bit behind him and rocket jumps off of the resulting explosion (jump/crouch). When he's in the air, he uses strafe to subtly change direction and keep his eyes on the soldier. Then reflects a rocket back at him, lands, then reflects again. The last reflect is an air shot.

Consider how fast the button presses and aim is to make that happen. Games like this fail on console because they don't work with a controller, the controller can't do what the game allows. The pyro in the video wouldn't even be anywhere near the ability of a top level TF2 player. He's just a good player, not a great one.

That's the difference between a controller and a keyboard.