Splatoon. ^^

Personally I think that has more to do with movement than shooting, but I agree, the Quake III engine just feels awesome. I don't play a lot of shooters in general, but I don't think I've ever played anything that feels as good as anything I've played that uses the Quake III engine. I think I've read John Carmack say that they put a lot of effort into clipping stuff/making sure the world feels 'solid' enough, and I think that's also a big part of it.

A few of mine:

Resident Evil 4. Every gun has a feeling of impact while still feeling like you're just barely holding back overwhelming odds. Throw in pixel-perfect pointer aiming on the Wii version and it's shooting nirvana.

F.E.A.R. The combination of visceral sound design, animation, and slow motion turns firefights into a glorious ballet of destruction. Vaporizing an enemy with the shotgun in particular never gets old. A classic.

Sniper Elite 3/4. The Mortal-Kombat-esque X ray kill cam gives kills a magnificent payoff, while using gyro aim to fine tune each shot is immensely immersive and gratifying.

This too, spraying ink around was great fun, especially with gyro.

As far as satisfying goes, Rage 2 filled that satisfaction for me greatly.

Contra III

Gungrave/Gungrave GORE

Assault Suit Leynos

Assault Suits Valken

Raiden series

Ikaruga

Space Marine 1-2

Binary Domain

Turbo Overkill

Gears 1-4.

VanQuish

Doom 2016

Prodeus

Anything from Toaplan

MUSHA

Bulletstorm

BT in Titanfall 2

Rage 2

If I thought about it can name a bunch more FPS,side scrolling run n guns. Shmups. Too many for me think of off the top of my head.

^

Agree with many of these.

Most Metal Slug games.